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Integer triangle

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An integer triangle or integral triangle is a <a href="Triangle" title="Triangle">triangle</a> all of whose side lengths are <a href="Integer" title="Integer">integers</a>. A rational triangle is one whose side lengths are <a href="Rational_number" title="Rational number">rational numbers</a>; any rational triangle can be <a href="Uniform_scaling" class="mw-redirect" title="Uniform scaling">rescaled</a> by the <a href="Lowest_common_denominator" title="Lowest common denominator">lowest common denominator</a> of the sides to obtain a <a href="Similar_triangle" class="mw-redirect" title="Similar triangle">similar</a> integer triangle, so there is a close relationship between integer triangles and rational triangles.

Sometimes other definitions of the term rational triangle are used: Carmichael (1914) and Dickson (1920) use the term to mean a <a href="Heronian_triangle" title="Heronian triangle">Heronian triangle</a> (a triangle with integral or rational side lengths and area);<a href="#cite_note-1">[1]</a> Conway and Guy (1996) define a rational triangle as one with rational sides and rational <a href="Angle" title="Angle">angles</a> measured in degrees—the only such triangles are rational-sided <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a>.<a href="#cite_note-CG-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="General_properties_for_an_integer_triangle">General properties for an integer triangle</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Integer_triangles_with_given_perimeter">Integer triangles with given perimeter</h3></div>

Any triple of positive integers can serve as the side lengths of an integer triangle as long as it satisfies the <a href="Triangle_inequality" title="Triangle inequality">triangle inequality</a>: the longest side is shorter than the sum of the other two sides. Each such triple defines an integer triangle that is unique <a href="Up_to" title="Up to">up to</a> <a href="Congruence_(geometry)" title="Congruence (geometry)">congruence</a>. So the number of integer triangles (up to congruence) with <a href="Perimeter" title="Perimeter">perimeter</a> p is the number of <a href="Partition_(number_theory)" class="mw-redirect" title="Partition (number theory)">partitions</a> of p into three positive parts that satisfy the triangle inequality. This is the integer closest to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2}/48}">

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<annotation encoding="application/x-tex">{\displaystyle p^{2}/48}</annotation>

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</math><img src="./0cd336dbe80b0afd4a135edfea81de54f9c4fcb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.8ex; height:3.176ex;" alt="{\displaystyle p^{2}/48}" loading="lazy"> when p is <a href="Parity_(mathematics)" title="Parity (mathematics)">even</a> and to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p+3)^{2}/48}">

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<annotation encoding="application/x-tex">{\displaystyle (p+3)^{2}/48}</annotation>

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</math><img src="./d23a6e76f4b52320ec742790218b40d8ba813b8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.523ex; height:3.176ex;" alt="{\displaystyle (p+3)^{2}/48}" loading="lazy"> when p is <a href="Parity_(mathematics)" title="Parity (mathematics)">odd</a>.<a href="#cite_note-Jenkyns-3">[3]</a><a href="#cite_note-4">[4]</a> It also means that the number of integer triangles with even numbered perimeters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2n}">

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<annotation encoding="application/x-tex">{\displaystyle p=2n}</annotation>

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</math><img src="./a6fc9f18255f3cc2ba0151c425c797c0fbe2ab81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.914ex; height:2.509ex;" alt="{\displaystyle p=2n}" loading="lazy"> is the same as the number of integer triangles with odd numbered perimeters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2n-3.}">

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<annotation encoding="application/x-tex">{\displaystyle p=2n-3.}</annotation>

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</math><img src="./9f5855b59d5d1bd9973f6f9b5a1c4a8cec4112d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.564ex; height:2.509ex;" alt="{\displaystyle p=2n-3.}" loading="lazy"> Thus there is no integer triangle with perimeter 1, 2 or 4, one with perimeter 3, 5, 6 or 8, and two with perimeter 7 or 10. The <a href="Integer_sequence" title="Integer sequence">sequence</a> of the number of integer triangles with perimeter p, starting at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1,}">

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<annotation encoding="application/x-tex">{\displaystyle p=1,}</annotation>

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</math><img src="./30adc7e357aca4250bff4dca805343ba0fd23f49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p=1,}" loading="lazy"> is:

<dl><dd>0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8 ... (sequence <a href="https://oeis.org/A005044" class="extiw external" title="oeis:A005044">A005044</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl>

This is called <a href="Alcuin's_sequence" title="Alcuin's sequence">Alcuin's sequence</a>.

<div class="mw-heading mw-heading3"><h3 id="Integer_triangles_with_given_largest_side">Integer triangles with given largest side</h3></div>

The number of integer triangles (up to congruence) with given largest side c and integer triple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c)}">

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<annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation>

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</math><img src="./ae973a762a92b9cd3eafe7f283890ccfa9b887e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}" loading="lazy"> is the number of integer triples such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b>c}">

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<annotation encoding="application/x-tex">{\displaystyle a+b>c}</annotation>

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</math><img src="./7609665556910463a555efc9158e41cd5cf463cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.173ex; height:2.343ex;" alt="{\displaystyle a+b>c}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b\leq c.}">

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<annotation encoding="application/x-tex">{\displaystyle a\leq b\leq c.}</annotation>

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</math><img src="./c4ee50ec34eaa0f97261545bd2ca3b917fb38a77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.078ex; height:2.343ex;" alt="{\displaystyle a\leq b\leq c.}" loading="lazy"> This is the integer value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lceil {\tfrac {1}{2}}(c+1)\rceil \cdot \lfloor {\tfrac {1}{2}}(c+1)\rfloor .}">

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<annotation encoding="application/x-tex">{\displaystyle \lceil {\tfrac {1}{2}}(c+1)\rceil \cdot \lfloor {\tfrac {1}{2}}(c+1)\rfloor .}</annotation>

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</math><img src="./cf32b90ff15c3e1f3895e23d9a1074fe81a44418.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.41ex; height:3.509ex;" alt="{\displaystyle \lceil {\tfrac {1}{2}}(c+1)\rceil \cdot \lfloor {\tfrac {1}{2}}(c+1)\rfloor .}" loading="lazy"><a href="#cite_note-Jenkyns-3">[3]</a> Alternatively, for c even it is the double <a href="Triangular_number" title="Triangular number">triangular number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}c{\bigl (}{\tfrac {1}{2}}c+1{\bigr )}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}c{\bigl (}{\tfrac {1}{2}}c+1{\bigr )}}</annotation>

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</math><img src="./e2a37c68096aa55820022f60ed78f05e57b39863.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.463ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}c{\bigl (}{\tfrac {1}{2}}c+1{\bigr )}}" loading="lazy"> and for c odd it is the <a href="Square_number" title="Square number">square</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}(c+1).}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}(c+1).}</annotation>

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</math><img src="./c4bede448b9a6a28baa7dd567cf3edc2acab80c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.124ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}(c+1).}" loading="lazy"> It also means that the number of integer triangles with greatest side c exceeds the number of integer triangles with greatest side c − 2 by c. The sequence of the number of non-congruent integer triangles with largest side c, starting at c = 1, is:

<dl><dd>1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90 ... (sequence <a href="https://oeis.org/A002620" class="extiw external" title="oeis:A002620">A002620</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl>

The number of integer triangles (up to congruence) with given largest side c and integer triple (a, b, c) that lie on or within a semicircle of diameter c is the number of integer triples such that a + b > c , a2 + b2 ≤ c2 and a ≤ b ≤ c. This is also the number of integer sided <a href="Obtuse_triangle" class="mw-redirect" title="Obtuse triangle">obtuse</a> or <a href="Right_triangle" title="Right triangle">right</a> (non-<a href="Acute_triangle" class="mw-redirect" title="Acute triangle">acute</a>) triangles with largest side c. The sequence starting at c = 1, is:

<dl><dd>0, 0, 1, 1, 3, 4, 5, 7, 10, 13, 15, 17, 22, 25, 30, 33, 38, 42, 48 ... (sequence <a href="https://oeis.org/A236384" class="extiw external" title="oeis:A236384">A236384</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl>

Consequently, the difference between the two above sequences gives the number of acute integer sided triangles (up to congruence) with given largest side c. The sequence starting at c = 1, is:

<dl><dd>1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, 52 ... (sequence <a href="https://oeis.org/A247588" class="extiw external" title="oeis:A247588">A247588</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Area_of_an_integer_triangle">Area of an integer triangle</h3></div>

By <a href="Heron's_formula" title="Heron's formula">Heron's formula</a>, if T is the <a href="Area" title="Area">area</a> of a triangle whose sides have lengths a, b, and c then

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<annotation encoding="application/x-tex">{\displaystyle 4T={\sqrt {(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}}.}</annotation>

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</math><img src="./82fe62cafe27bc71096029b27353514867b6438e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.573ex; height:4.843ex;" alt="{\displaystyle 4T={\sqrt {(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}}.}" loading="lazy"></dd></dl>

Since all the terms under the <a href="Square_root" title="Square root">radical</a> on the right side of the formula are integers it follows that all integer triangles must have 16T2 an integer and T2 will be rational.

<div class="mw-heading mw-heading3"><h3 id="Angles_of_an_integer_triangle">Angles of an integer triangle</h3></div>

By the <a href="Law_of_cosines" title="Law of cosines">law of cosines</a>, every angle of an integer triangle has a rational <a href="Cosine" class="mw-redirect" title="Cosine">cosine</a>. Every angle of an integer right triangle also has rational <a href="Sine" class="mw-redirect" title="Sine">sine</a> (see <a href="Pythagorean_triple" title="Pythagorean triple">Pythagorean triple</a>).

If the angles of any triangle form an <a href="Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a> then one of its angles must be 60°.<a href="#cite_note-Zelator-5">[5]</a> For integer triangles the remaining angles must also have rational cosines and a method of generating such triangles is given below. However, apart from the trivial case of an equilateral triangle, there are no integer triangles whose angles form either a <a href="Geometric_progression" title="Geometric progression">geometric</a> or <a href="Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">harmonic progression</a>. This is because such angles have to be rational angles of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi p/q}">

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<annotation encoding="application/x-tex">{\displaystyle \pi p/q}</annotation>

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</math><img src="./b35450bf3995946923f1c9a618c3c691f823b79e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.733ex; height:2.843ex;" alt="{\displaystyle \pi p/q}" loading="lazy"> with rational <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p/q<1.}">

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<annotation encoding="application/x-tex">{\displaystyle 0<p/q<1.}</annotation>

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</math><img src="./2d7625a18407777e0a59f4b4038c22d8ff36630c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.57ex; height:2.843ex;" alt="{\displaystyle 0<p/q<1.}" loading="lazy"> But all the angles of integer triangles must have rational cosines and this will occur only when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q=1/3.}">

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<annotation encoding="application/x-tex">{\displaystyle p/q=1/3.}</annotation>

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</math><img src="./5a065988b01e3e9c00a8fc24ded59291e7152bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.724ex; height:2.843ex;" alt="{\displaystyle p/q=1/3.}" loading="lazy"><a href="#cite_note-6">[6]</a>: p.2 i.e. the integer triangle is equilateral.

The square of each internal <a href="Angle_bisector" class="mw-redirect" title="Angle bisector">angle bisector</a> of an integer triangle is rational, because the general triangle formula for the internal angle bisector of angle A is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2{\sqrt {bcs(s-a)}}{\big /}(b+c)}">

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<annotation encoding="application/x-tex">{\textstyle 2{\sqrt {bcs(s-a)}}{\big /}(b+c)}</annotation>

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</math><img src="./8202fa150df82aeec397f1745712221e1a361448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.549ex; height:3.343ex;" alt="{\textstyle 2{\sqrt {bcs(s-a)}}{\big /}(b+c)}" loading="lazy"> where s is the <a href="Semiperimeter" title="Semiperimeter">semiperimeter</a> (and likewise for the other angles' bisectors).

<div class="mw-heading mw-heading3"><h3 id="Side_split_by_an_altitude">Side split by an altitude</h3></div>

Any <a href="Altitude_(triangle)" title="Altitude (triangle)">altitude</a> dropped from a vertex onto an opposite side or its extension will split that side or its extension into rational lengths.

<div class="mw-heading mw-heading3"><h3 id="Medians">Medians</h3></div>

The square of twice any <a href="Median_(geometry)" title="Median (geometry)">median</a> of an integer triangle is an integer, because the general formula for the squared median ma2 to side a is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}(2b^{2}+2c^{2}-a^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}(2b^{2}+2c^{2}-a^{2})}</annotation>

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</math><img src="./66c1ae4fcac928684aef302a2a397158e88c5a43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.87ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}(2b^{2}+2c^{2}-a^{2})}" loading="lazy">, giving (2ma)2 = 2b2 + 2c2 − a2 (and likewise for the medians to the other sides).

<div class="mw-heading mw-heading3"><h3 id="Circumradius_and_inradius">Circumradius and inradius</h3></div>

Because the square of the area of an integer triangle is rational, the square of its <a href="Circumradius" class="mw-redirect" title="Circumradius">circumradius</a> is also rational, as is the square of the <a href="Inradius" class="mw-redirect" title="Inradius">inradius</a>.

The ratio of the inradius to the circumradius of an integer triangle is rational, equaling <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4T^{2}/sabc}">

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<annotation encoding="application/x-tex">{\displaystyle 4T^{2}/sabc}</annotation>

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</math><img src="./27a4c3e9414640bb032cd456d80e74f5ca1b3f40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.424ex; height:3.176ex;" alt="{\displaystyle 4T^{2}/sabc}" loading="lazy"> for semiperimeter s and area T.

The product of the inradius and the circumradius of an integer triangle is rational, equaling <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle abc{\big /}2(a+b+c).}">

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<annotation encoding="application/x-tex">{\displaystyle abc{\big /}2(a+b+c).}</annotation>

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</math><img src="./bca3339aa4684b9cd4d3f52a64b57cc0d4e39021.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.111ex; height:3.176ex;" alt="{\displaystyle abc{\big /}2(a+b+c).}" loading="lazy">

Thus the squared distance between the <a href="Incenter" title="Incenter">incenter</a> and the <a href="Circumcenter" class="mw-redirect" title="Circumcenter">circumcenter</a> of an integer triangle, given by <a href="Euler's_theorem_in_geometry" title="Euler's theorem in geometry">Euler's theorem</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{2}-2Rr}">

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<annotation encoding="application/x-tex">{\displaystyle R^{2}-2Rr}</annotation>

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</math><img src="./6666822ea17a4d2a9877f6cb43ce00835970af1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.634ex; height:2.843ex;" alt="{\displaystyle R^{2}-2Rr}" loading="lazy"> is rational.

<div class="mw-heading mw-heading2"><h2 id="Heronian_triangles">Heronian triangles</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Heronian_triangle" title="Heronian triangle">Heronian triangle</a></div>

A Heronian triangle, also known as a Heron triangle or a Hero triangle, is a triangle with integer sides and integer area.

All Heronian triangles can be placed on a <a href="Lattice_(group)" title="Lattice (group)">lattice</a> with each vertex at a lattice point.<a href="#cite_note-7">[7]</a> Furthermore, if an integer triangle can be place on a lattice with each vertex at a lattice point it must be Heronian.

<div class="mw-heading mw-heading3"><h3 id="General_formula">General formula</h3></div>

Every Heronian triangle has sides proportional to<a href="#cite_note-8">[8]</a>

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<annotation encoding="application/x-tex">{\displaystyle a=n(m^{2}+k^{2})}</annotation>

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</math><img src="./48196ae962002021ed99d8280de0df142b8414b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.733ex; height:3.176ex;" alt="{\displaystyle a=n(m^{2}+k^{2})}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle b=m(n^{2}+k^{2})}</annotation>

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</math><img src="./3ecdb356601f425badf533fedb6e3c470323c4af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.501ex; height:3.176ex;" alt="{\displaystyle b=m(n^{2}+k^{2})}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle c=(m+n)(mn-k^{2})}</annotation>

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</math><img src="./26dd41f2d3e4e5acc7dacf0ad94780e359b49dc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.54ex; height:3.176ex;" alt="{\displaystyle c=(m+n)(mn-k^{2})}" loading="lazy"></dd>

<mtext>Semiperimeter</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Semiperimeter}}=mn(m+n)}</annotation>

</semantics>

</math><img src="./e098e2eed9cc90056e7de9525ccb80d7a4aa19ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.226ex; height:2.843ex;" alt="{\displaystyle {\text{Semiperimeter}}=mn(m+n)}" loading="lazy"></dd>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Area}}=mnk(m+n)(mn-k^{2})}</annotation>

</semantics>

</math><img src="./58b00869271d7e336c505577c5085e56ae287c7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.029ex; height:3.176ex;" alt="{\displaystyle {\text{Area}}=mnk(m+n)(mn-k^{2})}" loading="lazy"></dd></dl>

for integers m, n and k subject to the constraints:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gcd {(m,n,k)}=1}</annotation>

</semantics>

</math><img src="./b1b81836ff88bdc8c99da97ad98c18e24c027e45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.659ex; height:2.843ex;" alt="{\displaystyle \gcd {(m,n,k)}=1}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle mn>k^{2}\geq m^{2}n/(2m+n)}</annotation>

</semantics>

</math><img src="./2086006c8e7360480c1a5f5d1b9ba17e1db2fb86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.797ex; height:3.176ex;" alt="{\displaystyle mn>k^{2}\geq m^{2}n/(2m+n)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\geq n\geq 1.}</annotation>

</semantics>

</math><img src="./cb21d65b489f8740f4edce4a987092b37968e784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.441ex; height:2.343ex;" alt="{\displaystyle m\geq n\geq 1.}" loading="lazy"></dd></dl>

The proportionality factor is generally a rational <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p/q}</annotation>

</semantics>

</math><img src="./8fa5bd4cf049744deac0ac4a04c07998bd6befa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:3.491ex; height:2.843ex;" alt="{\displaystyle p/q}" loading="lazy"> where q = <a href="Greatest_common_divisor" title="Greatest common divisor">gcd</a>(a,b,c) reduces the generated Heronian triangle to its primitive and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> scales up this primitive to the required size.

<div class="mw-heading mw-heading3"><h3 id="Pythagorean_triangles">Pythagorean triangles</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Pythagorean_triple" title="Pythagorean triple">Pythagorean triple</a></div>

A Pythagorean triangle is right-angled and Heronian. Its three integer sides are known as a <a href="Pythagorean_triple" title="Pythagorean triple">Pythagorean triple</a> or Pythagorean triplet or Pythagorean triad.<a href="#cite_note-Sierpinski-9">[9]</a> All Pythagorean triples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation>

</semantics>

</math><img src="./ae973a762a92b9cd3eafe7f283890ccfa9b887e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}" loading="lazy"> with <a href="Hypotenuse" title="Hypotenuse">hypotenuse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> which are primitive (the sides having no <a href="Common_factor" class="mw-redirect" title="Common factor">common factor</a>) can be generated by

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=m^{2}-n^{2},\,}</annotation>

</semantics>

</math><img src="./c519bf2593bff91ac2e6e750df53e721dad2eab2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.746ex; height:3.009ex;" alt="{\displaystyle a=m^{2}-n^{2},\,}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=2mn,\,}</annotation>

</semantics>

</math><img src="./b6e2814b9823d159e962edbec7948012a960d56b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.728ex; height:2.509ex;" alt="{\displaystyle b=2mn,\,}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=m^{2}+n^{2},\,}</annotation>

</semantics>

</math><img src="./2f42232228dd937a4b18662470ac4988078024a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.523ex; height:3.009ex;" alt="{\displaystyle c=m^{2}+n^{2},\,}" loading="lazy"></dd>

<mtext>Semiperimeter</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Semiperimeter}}=m(m+n)\,}</annotation>

</semantics>

</math><img src="./ccbcb6b73648aaa4e8286aa2fd5c4cf69075eadb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.219ex; height:2.843ex;" alt="{\displaystyle {\text{Semiperimeter}}=m(m+n)\,}" loading="lazy"></dd>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Area}}=mn(m^{2}-n^{2})\,}</annotation>

</semantics>

</math><img src="./cfd42b353d3f9ab8d6d4796a488b1d020caf5afc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.964ex; height:3.176ex;" alt="{\displaystyle {\text{Area}}=mn(m^{2}-n^{2})\,}" loading="lazy"></dd></dl>

where m and n are <a href="Coprime" class="mw-redirect" title="Coprime">coprime</a> integers and one of them is even with m > n.

Every even number greater than 2 can be the leg of a Pythagorean triangle (not necessarily primitive) because if the leg is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=2m}</annotation>

</semantics>

</math><img src="./04fb62f2ef7837793c038be5f4d289e0de66671d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.531ex; height:2.176ex;" alt="{\displaystyle a=2m}" loading="lazy"> and we choose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=(a/2)^{2}-1=m^{2}-1}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=(a/2)^{2}-1=m^{2}-1}</annotation>

</semantics>

</math><img src="./bf30380c114b54ea2464cd18f5443ea4be136089.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.713ex; height:3.176ex;" alt="{\displaystyle b=(a/2)^{2}-1=m^{2}-1}" loading="lazy"> as the other leg then the hypotenuse is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=m^{2}+1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=m^{2}+1}</annotation>

</semantics>

</math><img src="./6a2cd018f17aef99344141d0450297625937b683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.203ex; height:2.843ex;" alt="{\displaystyle c=m^{2}+1}" loading="lazy">.<a href="#cite_note-pyth_tr_area-10">[10]</a> This is essentially the generation formula above with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> set to 1 and allowing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> to range from 2 to infinity.

<div class="mw-heading mw-heading4"><h4 id="Pythagorean_triangles_with_integer_altitude_from_the_hypotenuse">Pythagorean triangles with integer altitude from the hypotenuse</h4></div>

There are no primitive Pythagorean triangles with integer altitude from the hypotenuse. This is because twice the area equals any base times the corresponding height: 2 times the area thus equals both ab and cd where d is the height from the hypotenuse c. The three side lengths of a primitive triangle are coprime, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=ab/c}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=ab/c}</annotation>

</semantics>

</math><img src="./8ec0118e8d8c451d2ae375fadfaa0c5172dec8ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.711ex; height:2.843ex;" alt="{\displaystyle d=ab/c}" loading="lazy"> is in fully reduced form; since c cannot equal 1 for any primitive Pythagorean triangle, d cannot be an integer.

However, any Pythagorean triangle with legs x, y and hypotenuse z can generate a Pythagorean triangle with an integer altitude, by scaling up the sides by the length of the hypotenuse z. If d is the altitude, then the generated Pythagorean triangle with integer altitude is given by<a href="#cite_note-Richinik-11">[11]</a>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle (a,b,c,d)=(xz,yz,z^{2},xy).\,}</annotation>

</semantics>

</math><img src="./8cb6d2db69c11172a9b56c11dc604a3c52868a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.696ex; height:3.176ex;" alt="{\displaystyle (a,b,c,d)=(xz,yz,z^{2},xy).\,}" loading="lazy"></dd></dl>

Consequently, all Pythagorean triangles with legs a and b, hypotenuse c, and integer altitude d from the hypotenuse, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(a,b,c,d)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gcd(a,b,c,d)=1}</annotation>

</semantics>

</math><img src="./e002a6b66f1cb727d7356aeef6413011d19e8190.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.109ex; height:2.843ex;" alt="{\displaystyle \gcd(a,b,c,d)=1}" loading="lazy">, which necessarily satisfy both a2 + b2 = c2 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{a^{2}}}+{\tfrac {1}{b^{2}}}={\tfrac {1}{d^{2}}}}">

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

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</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{a^{2}}}+{\tfrac {1}{b^{2}}}={\tfrac {1}{d^{2}}}}</annotation>

</semantics>

</math><img src="./6cef94305be88de640865db73c591f8e7464677a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.378ex; height:4.009ex;" alt="{\displaystyle {\tfrac {1}{a^{2}}}+{\tfrac {1}{b^{2}}}={\tfrac {1}{d^{2}}}}" loading="lazy">, are generated by<a href="#cite_note-12">[12]</a><a href="#cite_note-Richinik-11">[11]</a>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=(m^{2}-n^{2})(m^{2}+n^{2}),\,}</annotation>

</semantics>

</math><img src="./6785d5d0ce0373575297eec7938e2ed9b0870297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.749ex; height:3.176ex;" alt="{\displaystyle a=(m^{2}-n^{2})(m^{2}+n^{2}),\,}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=2mn(m^{2}+n^{2}),\,}</annotation>

</semantics>

</math><img src="./60fda6541f13f00e4e07704956f1e899490a532d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.921ex; height:3.176ex;" alt="{\displaystyle b=2mn(m^{2}+n^{2}),\,}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=(m^{2}+n^{2})^{2},\,}</annotation>

</semantics>

</math><img src="./2e821ced18e92a76772c66096f8f56810f5558ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.387ex; height:3.176ex;" alt="{\displaystyle c=(m^{2}+n^{2})^{2},\,}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=2mn(m^{2}-n^{2}),\,}</annotation>

</semantics>

</math><img src="./6a34f3bce484eddcdf260d8bb8eed583cccd6919.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.139ex; height:3.176ex;" alt="{\displaystyle d=2mn(m^{2}-n^{2}),\,}" loading="lazy"></dd></dl>

<mtext>Semiperimeter</mtext>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Semiperimeter}}=m(m+n)(m^{2}+n^{2})\,}</annotation>

</semantics>

</math><img src="./30435282da983cb5e2c5e5ff665054bf30ed003d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.412ex; height:3.176ex;" alt="{\displaystyle {\text{Semiperimeter}}=m(m+n)(m^{2}+n^{2})\,}" loading="lazy"></dd></dl>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Area}}=mn(m^{2}-n^{2})(m^{2}+n^{2})^{2}\,}</annotation>

</semantics>

</math><img src="./5f4c777a101ae6b170183318f5454d525d553b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.211ex; height:3.176ex;" alt="{\displaystyle {\text{Area}}=mn(m^{2}-n^{2})(m^{2}+n^{2})^{2}\,}" loading="lazy"></dd></dl>

for coprime integers m, n with m > n.

<div class="mw-heading mw-heading3"><h3 id="Heronian_triangles_with_sides_in_arithmetic_progression">Heronian triangles with sides in arithmetic progression</h3></div>

A triangle with integer sides and integer area has sides in arithmetic progression <a href="If_and_only_if" title="If and only if">if and only if</a><a href="#cite_note-Buchholz-13">[13]</a> the sides are (b – d, b, b + d), where

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=2(m^{2}+3n^{2})/g,}</annotation>

</semantics>

</math><img src="./3365c540e0ae61ae373a9fd5416dacf9c43e5ec6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.54ex; height:3.176ex;" alt="{\displaystyle b=2(m^{2}+3n^{2})/g,}" loading="lazy"></dd></dl>

<msup>

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</msup>

<msup>

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</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=(m^{2}-3n^{2})/g,}</annotation>

</semantics>

</math><img src="./f2bc2b7b3d6d50b04bf8ee4a787e29b3ee4e6f66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.595ex; height:3.176ex;" alt="{\displaystyle d=(m^{2}-3n^{2})/g,}" loading="lazy"></dd></dl>

and where g is the <a href="Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m^{2}-3n^{2},}">

<msup>

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</msup>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle m^{2}-3n^{2},}</annotation>

</semantics>

</math><img src="./60c6683309a43ab128dbf0625908f6658aa9ae91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.193ex; height:3.009ex;" alt="{\displaystyle m^{2}-3n^{2},}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2mn,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2mn,}</annotation>

</semantics>

</math><img src="./3e9c10f98606100e55dc003449a7837d0190a4a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.244ex; height:2.509ex;" alt="{\displaystyle 2mn,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m^{2}+3n^{2}.}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m^{2}+3n^{2}.}</annotation>

</semantics>

</math><img src="./e68307d498d69bdab7ffa80c157b7feb459f6a69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.193ex; height:2.843ex;" alt="{\displaystyle m^{2}+3n^{2}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Heronian_triangles_with_one_angle_equal_to_twice_another">Heronian triangles with one angle equal to twice another</h3></div>

All Heronian triangles with B = 2A are generated by<a href="#cite_note-14">[14]</a> either

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<mtr>

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<mtd>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&={\tfrac {1}{4}}k^{2}(s^{2}+r^{2})^{2},\\[5mu]b&={\tfrac {1}{2}}k^{2}(s^{4}-r^{4}),\\[5mu]c&={\tfrac {1}{4}}k^{2}(3s^{4}-10s^{2}r^{2}+3r^{4}),\\[5mu]{\text{Area}}&={\tfrac {1}{2}}k^{2}csr(s^{2}-r^{2}),\end{aligned}}}</annotation>

</semantics>

</math><img src="./737789b2c22c27725965ab7b98bac198a9dcfb4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.647ex; margin-bottom: -0.191ex; width:33.905ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}a&={\tfrac {1}{4}}k^{2}(s^{2}+r^{2})^{2},\\[5mu]b&={\tfrac {1}{2}}k^{2}(s^{4}-r^{4}),\\[5mu]c&={\tfrac {1}{4}}k^{2}(3s^{4}-10s^{2}r^{2}+3r^{4}),\\[5mu]{\text{Area}}&={\tfrac {1}{2}}k^{2}csr(s^{2}-r^{2}),\end{aligned}}}" loading="lazy"></dd></dl>

with integers k, s, r such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}>3r^{2},}">

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle s^{2}>3r^{2},}</annotation>

</semantics>

</math><img src="./b6bc24cccd0eb41f29b409bd5facac475cf0c6ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.155ex; height:3.009ex;" alt="{\displaystyle s^{2}>3r^{2},}" loading="lazy"> or

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<mtr>

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<mtd>

<mfrac>

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<mtd>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&={\tfrac {1}{4}}q^{2}(u^{2}+v^{2})^{2},\\[5mu]b&=q^{2}uv(u^{2}+v^{2}),\\[5mu]c&={\tfrac {1}{4}}q^{2}(14u^{2}v^{2}-u^{4}-v^{4}),\\[5mu]{\text{Area}}&={\tfrac {1}{2}}q^{2}cuv(v^{2}-u^{2}),\end{aligned}}}</annotation>

</semantics>

</math><img src="./0d37943afde41e546850d91dbd1d98d8af0f4f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:32.085ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}a&={\tfrac {1}{4}}q^{2}(u^{2}+v^{2})^{2},\\[5mu]b&=q^{2}uv(u^{2}+v^{2}),\\[5mu]c&={\tfrac {1}{4}}q^{2}(14u^{2}v^{2}-u^{4}-v^{4}),\\[5mu]{\text{Area}}&={\tfrac {1}{2}}q^{2}cuv(v^{2}-u^{2}),\end{aligned}}}" loading="lazy"></dd></dl>

with integers q, u, v such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v>u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v>u}</annotation>

</semantics>

</math><img src="./dc20c9ec37be75fbf6c34a52fa345e96c637311c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.556ex; height:1.843ex;" alt="{\displaystyle v>u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}<(7+4{\sqrt {3}})u^{2}.}">

<msup>

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</msup>

<msqrt>

</msqrt>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle v^{2}<(7+4{\sqrt {3}})u^{2}.}</annotation>

</semantics>

</math><img src="./0ba213d453db1cb06510f720342eabb96c5dc9ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.384ex; height:3.176ex;" alt="{\displaystyle v^{2}<(7+4{\sqrt {3}})u^{2}.}" loading="lazy">

No Heronian triangles with B = 2A are isosceles or right triangles because all resulting angle combinations generate angles with non-rational <a href="Sine" class="mw-redirect" title="Sine">sines</a>, giving a non-rational area or side.

<div class="mw-heading mw-heading3"><h3 id="Isosceles_Heronian_triangles">Isosceles Heronian triangles</h3></div>

All <a href="Isosceles_triangle" title="Isosceles triangle">isosceles</a> Heronian triangles are decomposable. They are formed by joining two congruent Pythagorean triangles along either of their common legs such that the equal sides of the isosceles triangle are the hypotenuses of the Pythagorean triangles, and the base of the isosceles triangle is twice the other Pythagorean leg. Consequently, every Pythagorean triangle is the building block for two isosceles Heronian triangles since the join can be along either leg.

All pairs of isosceles Heronian triangles are given by rational multiples of the following side lengths:<a href="#cite_note-Sastry-15">[15]</a>

<mtr>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&=2(u^{2}-v^{2}),&\quad \quad a&=4uv,\\b&=u^{2}+v^{2},&\quad \quad b&=u^{2}+v^{2},\\c&=u^{2}+v^{2},&\quad \quad c&=u^{2}+v^{2},\\{\text{Area}}&=2uv(u^{2}-v^{2}),\end{aligned}}}</annotation>

</semantics>

</math><img src="./e892d1b3a43fa57cba02effdb8b2b77deb5259c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:43.853ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}a&=2(u^{2}-v^{2}),&\quad \quad a&=4uv,\\b&=u^{2}+v^{2},&\quad \quad b&=u^{2}+v^{2},\\c&=u^{2}+v^{2},&\quad \quad c&=u^{2}+v^{2},\\{\text{Area}}&=2uv(u^{2}-v^{2}),\end{aligned}}}" loading="lazy">

for coprime integers of opposite parity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u>v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u>v}</annotation>

</semantics>

</math><img src="./e586068b93531f3d8481d38585fd34a72620e946.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.556ex; height:1.843ex;" alt="{\displaystyle u>v}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Heronian_triangles_whose_perimeter_is_four_times_a_prime">Heronian triangles whose perimeter is four times a prime</h3></div>

It has been shown that a Heronian triangle whose perimeter is four times a <a href="Prime_number" title="Prime number">prime</a> is uniquely associated with the prime and that the prime is <a href="Modular_arithmetic" title="Modular arithmetic">congruent</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3}</annotation>

</semantics>

</math><img src="./991e33c6e207b12546f15bdfee8b5726eafbbb2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 8}</annotation>

</semantics>

</math><img src="./1aaa997e6ad67716cfaa9a02c4df860bf60a95b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 8}" loading="lazy">.<a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a> It is well known that such a prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> can be uniquely partitioned into integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=m^{2}+2n^{2}}">

<msup>

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</msup>

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=m^{2}+2n^{2}}</annotation>

</semantics>

</math><img src="./3960ea139d62e39fddf71ecc8070339f682fe931.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.904ex; height:3.009ex;" alt="{\displaystyle p=m^{2}+2n^{2}}" loading="lazy"> (see <a href="Idoneal_number" title="Idoneal number">Euler's idoneal numbers</a>). Furthermore, it has been shown that such Heronian triangles are primitive since the smallest side of the triangle has to be equal to the prime that is one quarter of its perimeter.

Consequently, all primitive Heronian triangles whose perimeter is four times a prime can be generated by

<msup>

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<annotation encoding="application/x-tex">{\displaystyle a=m^{2}+2n^{2}}</annotation>

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</math><img src="./3cf2a11fe0d918a54472b59d77e61abae091af81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.875ex; height:2.843ex;" alt="{\displaystyle a=m^{2}+2n^{2}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle b=m^{2}+4n^{2}}</annotation>

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</math><img src="./af92dba8e23fba637cc018d555c6936a0f5bd930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.642ex; height:2.843ex;" alt="{\displaystyle b=m^{2}+4n^{2}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle c=2(m^{2}+n^{2})}</annotation>

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</math><img src="./e2561c4984a146eb6a2cc1928d17d29d6a7cfc97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.461ex; height:3.176ex;" alt="{\displaystyle c=2(m^{2}+n^{2})}" loading="lazy"></dd>

<mtext>Semiperimeter</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Semiperimeter}}=2a=2(m^{2}+2n^{2})}</annotation>

</semantics>

</math><img src="./9381236d5532f18dcd92fa4c62b2efd2749891d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.715ex; height:3.176ex;" alt="{\displaystyle {\text{Semiperimeter}}=2a=2(m^{2}+2n^{2})}" loading="lazy"></dd>

</mrow>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Area}}=2mn(m^{2}+2n^{2})}</annotation>

</semantics>

</math><img src="./41cc7a4a663819f458d0a1a1b8d270cac56141a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.901ex; height:3.176ex;" alt="{\displaystyle {\text{Area}}=2mn(m^{2}+2n^{2})}" loading="lazy"></dd></dl>

for integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m^{2}+2n^{2}}">

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle m^{2}+2n^{2}}</annotation>

</semantics>

</math><img src="./34976f04f8f7abe27e7db23646b9e3b2612d1142.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.546ex; height:2.843ex;" alt="{\displaystyle m^{2}+2n^{2}}" loading="lazy"> is a prime.

Furthermore, the factorization of the area is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2mnp}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 2mnp}</annotation>

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</math><img src="./de6e7bf56bfd0b33979a8b7fc70e0a33f05992fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.767ex; height:2.509ex;" alt="{\displaystyle 2mnp}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=m^{2}+2n^{2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle p=m^{2}+2n^{2}}</annotation>

</semantics>

</math><img src="./3960ea139d62e39fddf71ecc8070339f682fe931.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.904ex; height:3.009ex;" alt="{\displaystyle p=m^{2}+2n^{2}}" loading="lazy"> is prime. However the area of a Heronian triangle is always divisible by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6}">

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<annotation encoding="application/x-tex">{\displaystyle 6}</annotation>

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</math><img src="./39d81124420a058a7474dfeda48228fb6ee1e253.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}" loading="lazy">. This gives the result that apart from when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=1}</annotation>

</semantics>

</math><img src="./b6100c5ebd48c6fd848709f2be624465203eb173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1,}</annotation>

</semantics>

</math><img src="./dba4db03e5186e479ecd9611484b8657140a7ff0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.302ex; height:2.509ex;" alt="{\displaystyle n=1,}" loading="lazy"> which gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=3,}">

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<annotation encoding="application/x-tex">{\displaystyle p=3,}</annotation>

</semantics>

</math><img src="./f54aabf9cb1810df7de7343a7f33e17b48bb94f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p=3,}" loading="lazy"> all other parings of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> must have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> odd with only one of them divisible by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3}</annotation>

</semantics>

</math><img src="./991e33c6e207b12546f15bdfee8b5726eafbbb2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Heronian_triangles_with_rational_angle_bisectors">Heronian triangles with rational angle bisectors</h3></div>

If in a Heronian triangle the angle bisector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{a}}</annotation>

</semantics>

</math><img src="./6722f96d8470ee21d1dcf4f392684a02ce64c8b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.766ex; height:2.009ex;" alt="{\displaystyle w_{a}}" loading="lazy"> of the angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">, the angle bisector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{b}}</annotation>

</semantics>

</math><img src="./ace38d24b2a108a2a55dc7800626d4c1ea9cd4de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.602ex; height:2.009ex;" alt="{\displaystyle w_{b}}" loading="lazy"> of the angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> and the angle bisector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{c}}</annotation>

</semantics>

</math><img src="./9868b2118a7dfc0a2967ebb725651cc0f1b77c09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.608ex; height:2.009ex;" alt="{\displaystyle w_{c}}" loading="lazy"> of the angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> have a rational relationship with the three sides then not only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma }</annotation>

</semantics>

</math><img src="./301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.15ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma }" loading="lazy"> but also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\alpha }">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\alpha }</annotation>

</semantics>

</math><img src="./8ad6844ceafa45ecd24f6f0133ac827714e0b49e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.146ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\alpha }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\beta }">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\beta }</annotation>

</semantics>

</math><img src="./a4b6550456f3d063bc260863265900fe58427d0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.99ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\beta }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\gamma }">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\gamma }</annotation>

</semantics>

</math><img src="./f5575b01614b87750de891d9442c23a2e34831c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.92ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\gamma }" loading="lazy"> must be Heronian angles. Namely, if both angles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\alpha }">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\alpha }</annotation>

</semantics>

</math><img src="./8ad6844ceafa45ecd24f6f0133ac827714e0b49e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.146ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\alpha }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\beta }">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\beta }</annotation>

</semantics>

</math><img src="./a4b6550456f3d063bc260863265900fe58427d0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.99ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\beta }" loading="lazy"> are Heronian then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\gamma }">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\gamma }</annotation>

</semantics>

</math><img src="./f5575b01614b87750de891d9442c23a2e34831c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.92ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\gamma }" loading="lazy">, the complement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\alpha +{\tfrac {1}{2}}\beta }">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\alpha +{\tfrac {1}{2}}\beta }</annotation>

</semantics>

</math><img src="./113c69acef3810aa9a6d5a619944a39eda582220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.976ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\alpha +{\tfrac {1}{2}}\beta }" loading="lazy">, must also be a Heronian angle, so that all three angle-bisectors are rational. This is also evident if one multiplies:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{a}={\frac {2{\sqrt {s(s-a)}}\cdot {\sqrt {bc}}}{b+c}}\quad w_{b}={\frac {2{\sqrt {s(s-b)}}\cdot {\sqrt {ac}}}{a+c}}\quad w_{c}={\frac {2{\sqrt {s(s-c)}}\cdot {\sqrt {ab}}}{a+b}}}</annotation>

</semantics>

</math><img src="./1a41d4591a147a786e7fecf92a4c3adcba65f0c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:75.923ex; height:6.509ex;" alt="{\displaystyle w_{a}={\frac {2{\sqrt {s(s-a)}}\cdot {\sqrt {bc}}}{b+c}}\quad w_{b}={\frac {2{\sqrt {s(s-b)}}\cdot {\sqrt {ac}}}{a+c}}\quad w_{c}={\frac {2{\sqrt {s(s-c)}}\cdot {\sqrt {ab}}}{a+b}}}" loading="lazy"></dd></dl>

together. Namely, through this one obtains:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{a}\cdot w_{b}\cdot w_{c}={\frac {8s\cdot J\cdot a\cdot b\cdot c}{(a+b)(a+c)(b+c)}},}</annotation>

</semantics>

</math><img src="./9320f45344bc42b89978c4929a4636f8e041d3ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.333ex; height:6.176ex;" alt="{\displaystyle w_{a}\cdot w_{b}\cdot w_{c}={\frac {8s\cdot J\cdot a\cdot b\cdot c}{(a+b)(a+c)(b+c)}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> denotes the semi-perimeter, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

</semantics>

</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> the area of the triangle.

All similarity classes of Heronian triangles with rational angle bisectors are generated by<a href="#cite_note-18">[18]</a>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=mn(p^{2}+q^{2})}</annotation>

</semantics>

</math><img src="./b61d4b5c7d8113d10161b1db0009b1ea2df2116f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.77ex; height:3.176ex;" alt="{\displaystyle a=mn(p^{2}+q^{2})}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=pq(m^{2}+n^{2})}</annotation>

</semantics>

</math><img src="./9bc2c3e72794e75a44970aa452c39209ebe3678f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.528ex; height:3.176ex;" alt="{\displaystyle b=pq(m^{2}+n^{2})}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=(mq+np)(mp-nq)}</annotation>

</semantics>

</math><img src="./ac71f11437b5c7745e13bb8fcc50e8fb9d4994bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.753ex; height:2.843ex;" alt="{\displaystyle c=(mq+np)(mp-nq)}" loading="lazy"></dd>

<mtext>Semiperimeter</mtext>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Semiperimeter}}=s=(a+b+c)/2=mp(mq+np)}</annotation>

</semantics>

</math><img src="./490f5172bc6644ba611ef55953f5b8921ecb0d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.577ex; height:2.843ex;" alt="{\displaystyle {\text{Semiperimeter}}=s=(a+b+c)/2=mp(mq+np)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s-a=mq(mp-nq)}</annotation>

</semantics>

</math><img src="./1e894a6d624c7f86f7cf348c87ea1ddb7f6befbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.693ex; height:2.843ex;" alt="{\displaystyle s-a=mq(mp-nq)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s-b=np(mp-nq)}</annotation>

</semantics>

</math><img src="./fa48588a8c8e033cd49c5067f473f15a8165e8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.915ex; height:2.843ex;" alt="{\displaystyle s-b=np(mp-nq)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s-c=nq(mq+np)}</annotation>

</semantics>

</math><img src="./3c072f58d28371bac95ae13d6405fc35963d67c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.824ex; height:2.843ex;" alt="{\displaystyle s-c=nq(mq+np)}" loading="lazy"></dd>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Area}}=J=mnpq(mq+np)(mp-nq)}</annotation>

</semantics>

</math><img src="./c1cb970b12749cf4361edff931d84e527de39714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.839ex; height:2.843ex;" alt="{\displaystyle {\text{Area}}=J=mnpq(mq+np)(mp-nq)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n,p,q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,n,p,q}</annotation>

</semantics>

</math><img src="./20a70cc8e581c86a592b0bdc3251c2b343d98645.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.776ex; height:2.009ex;" alt="{\displaystyle m,n,p,q}" loading="lazy"> are such that

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=t^{2}-u^{2}}</annotation>

</semantics>

</math><img src="./a7d514e97ca922720e165413b077daff1635b050.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.257ex; height:2.843ex;" alt="{\displaystyle m=t^{2}-u^{2}}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2tu}</annotation>

</semantics>

</math><img src="./a9284ccfa3556c684b5e360e7362077512a0997f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.825ex; height:2.176ex;" alt="{\displaystyle n=2tu}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=v^{2}-w^{2}}</annotation>

</semantics>

</math><img src="./7b6fa85bd25e1783123d5fcb6bc8817dffab53d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.098ex; height:3.009ex;" alt="{\displaystyle p=v^{2}-w^{2}}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=2vw}</annotation>

</semantics>

</math><img src="./a4487497215b639e0aefe40efaf9973e3b67574a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.122ex; height:2.509ex;" alt="{\displaystyle q=2vw}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,u,v,w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t,u,v,w}</annotation>

</semantics>

</math><img src="./8936c57a6c1caea34476ed30c65bb06659b102e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.063ex; height:2.343ex;" alt="{\displaystyle t,u,v,w}" loading="lazy"> are arbitrary integers such that

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> coprime,</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> coprime.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Heronian_triangles_with_integer_inradius_and_exradii">Heronian triangles with integer inradius and exradii</h3></div>

There are infinitely many decomposable, and infinitely many indecomposable, primitive Heronian (non-Pythagorean) triangles with integer radii for the <a href="Incircle" class="mw-redirect" title="Incircle">incircle</a> and each <a href="Excircle" class="mw-redirect" title="Excircle">excircle</a>.<a href="#cite_note-Zhou-19">[19]</a>: Thms. 3 and 4 A family of decomposable ones is given by

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=4n^{2}}</annotation>

</semantics>

</math><img src="./ec7b71088b17bf09fbb8b2d6254e5be915d02316.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.94ex; height:2.676ex;" alt="{\displaystyle a=4n^{2}}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=(2n+1)(2n^{2}-2n+1)}</annotation>

</semantics>

</math><img src="./223a11caa00633da4355e21f307f50698412740c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.286ex; height:3.176ex;" alt="{\displaystyle b=(2n+1)(2n^{2}-2n+1)}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=(2n-1)(2n^{2}+2n+1)}</annotation>

</semantics>

</math><img src="./8c16b29a9443b4f9125ffa61ff09aa9c9c164355.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.296ex; height:3.176ex;" alt="{\displaystyle c=(2n-1)(2n^{2}+2n+1)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r=2n-1}</annotation>

</semantics>

</math><img src="./fea885e581d8bfcbc70d5d51b308c9f1937dae93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.707ex; height:2.343ex;" alt="{\displaystyle r=2n-1}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{a}=2n+1}</annotation>

</semantics>

</math><img src="./3b4f171f8fbca6c16378d9efdc39e36ae9165db0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.809ex; height:2.509ex;" alt="{\displaystyle r_{a}=2n+1}" loading="lazy"></dd>

<msub>

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</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{b}=2n^{2}}</annotation>

</semantics>

</math><img src="./db5834bc89d5569c7cca98fef1f034fcb6e81db8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.696ex; height:3.009ex;" alt="{\displaystyle r_{b}=2n^{2}}" loading="lazy"></dd>

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{c}={\text{Area}}=2n^{2}(2n-1)(2n+1);}</annotation>

</semantics>

</math><img src="./eae5705dc4cd263fde41e3689f598db1f84344ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.036ex; height:3.176ex;" alt="{\displaystyle r_{c}={\text{Area}}=2n^{2}(2n-1)(2n+1);}" loading="lazy"></dd></dl>

and a family of indecomposable ones is given by

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=5(5n^{2}+n-1)}</annotation>

</semantics>

</math><img src="./0719a24a022ca1be27478085c3b24b01aa5f6a76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.149ex; height:3.176ex;" alt="{\displaystyle a=5(5n^{2}+n-1)}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle b=(5n+3)(5n^{2}-4n+1)}</annotation>

</semantics>

</math><img src="./86e9c6e8de3d58252f6332ad845a188f2db41322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.286ex; height:3.176ex;" alt="{\displaystyle b=(5n+3)(5n^{2}-4n+1)}" loading="lazy"></dd>

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle c=(5n-2)(5n^{2}+6n+2)}</annotation>

</semantics>

</math><img src="./58ebaf158285a46683af685e3278c96bcd35c4e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.296ex; height:3.176ex;" alt="{\displaystyle c=(5n-2)(5n^{2}+6n+2)}" loading="lazy"></dd>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle r=5n-2}</annotation>

</semantics>

</math><img src="./465c8619a69d53c2f72433e3de6965783f6961c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.707ex; height:2.343ex;" alt="{\displaystyle r=5n-2}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle r_{a}=5n+3}</annotation>

</semantics>

</math><img src="./32b5cefac458eff8ca2b5209b7d927aedbb2aa99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.809ex; height:2.509ex;" alt="{\displaystyle r_{a}=5n+3}" loading="lazy"></dd>

<msub>

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</msub>

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle r_{b}=5n^{2}+n-1}</annotation>

</semantics>

</math><img src="./a037e6252ba57845fdc771dfa105527c1b5effb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.934ex; height:3.009ex;" alt="{\displaystyle r_{b}=5n^{2}+n-1}" loading="lazy"></dd>

<msub>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle r_{c}={\text{Area}}=(5n-2)(5n+3)(5n^{2}+n-1).}</annotation>

</semantics>

</math><img src="./f16d149e126796ba30d5888257a4c1e6380d8b9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.083ex; height:3.176ex;" alt="{\displaystyle r_{c}={\text{Area}}=(5n-2)(5n+3)(5n^{2}+n-1).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Heronian_triangles_as_faces_of_a_tetrahedron">Heronian triangles as faces of a tetrahedron</h3></div>

There exist <a href="Tetrahedra" class="mw-redirect" title="Tetrahedra">tetrahedra</a> having integer-valued <a href="Volume" title="Volume">volume</a> and Heron triangles as <a href="Face_(geometry)" title="Face (geometry)">faces</a>. One example has one edge of 896, the opposite edge of 190, and the other four edges of 1073; two faces have areas of 436800 and the other two have areas of 47120, while the volume is 62092800.<a href="#cite_note-Sierpinski-9">[9]</a>: p.107

<div class="mw-heading mw-heading3"><h3 id="Heronian_triangles_in_a_2D_lattice">Heronian triangles in a 2D lattice</h3></div>

A 2D <a href="Lattice_graph" title="Lattice graph">lattice</a> is a regular array of isolated points where if any one point is chosen as the <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian origin</a> (0, 0), then all the other points are at (x, y) where x and y range over all positive and negative integers. A lattice triangle is any triangle drawn within a 2D lattice such that all vertices lie on lattice points. By <a href="Pick's_theorem" title="Pick's theorem">Pick's theorem</a> a lattice triangle has a rational area that either is an integer or a <a href="Half-integer" title="Half-integer">half-integer</a> (has a denominator of 2). If the lattice triangle has integer sides then it is Heronian with integer area.<a href="#cite_note-Buchholz1-20">[20]</a>

Furthermore, it has been proved that all Heronian triangles can be drawn as lattice triangles.<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> Consequently, an integer triangle is Heronian if and only if it can be drawn as a lattice triangle.

There are infinitely many primitive Heronian (non-Pythagorean) triangles which can be placed on an <a href="Integer_lattice" title="Integer lattice">integer lattice</a> with all vertices, the <a href="Incenter" title="Incenter">incenter</a>, and all three <a href="Excenter" class="mw-redirect" title="Excenter">excenters</a> at lattice points. Two families of such triangles are the ones with parametrizations given above at <a href="#Heronian_triangles_with_integer_inradius_and_exradii">#Heronian triangles with integer inradius and exradii</a>.<a href="#cite_note-Zhou-19">[19]</a>: Thm. 5

<div class="mw-heading mw-heading2"><h2 id="Integer_automedian_triangles">Integer automedian triangles</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Automedian_triangle" title="Automedian triangle">Automedian triangle</a></div>

An <a href="Automedian_triangle" title="Automedian triangle">automedian triangle</a> is one whose medians are in the same proportions (in the opposite order) as the sides. If x, y, and z are the three sides of a right triangle, sorted in increasing order by size, and if 2x < z, then z, x + y, and y − x are the three sides of an automedian triangle. For instance, the right triangle with side lengths 5, 12, and 13 can be used in this way to form the smallest non-trivial (i.e., non-equilateral) integer automedian triangle, with side lengths 13, 17, and 7.<a href="#cite_note-parry-23">[23]</a>

Consequently, using <a href="Pythagorean_triple#Proof_of_Euclid.27s_formula" title="Pythagorean triple">Euclid's formula</a>, which generates primitive Pythagorean triangles, it is possible to generate primitive integer automedian triangles as

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<annotation encoding="application/x-tex">{\displaystyle a=|m^{2}-2mn-n^{2}|}</annotation>

</semantics>

</math><img src="./797ea497fa508d30965946003575a939cf129b43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.444ex; height:3.176ex;" alt="{\displaystyle a=|m^{2}-2mn-n^{2}|}" loading="lazy"></dd>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=m^{2}+2mn-n^{2}}</annotation>

</semantics>

</math><img src="./40d2079adb00528f2a86784149965ab2816428c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.918ex; height:2.843ex;" alt="{\displaystyle b=m^{2}+2mn-n^{2}}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=m^{2}+n^{2}}</annotation>

</semantics>

</math><img src="./cb4269272d3a068ea2d40f4620091aa78a9929f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.489ex; height:2.843ex;" alt="{\displaystyle c=m^{2}+n^{2}}" loading="lazy"></dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> coprime and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m+n}</annotation>

</semantics>

</math><img src="./88528fefcfac1b22d2df9b71d0f4fc9e758f65ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.176ex;" alt="{\displaystyle m+n}" loading="lazy"> odd, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<m<n{\sqrt {3}}}">

<msqrt>

</msqrt>

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<annotation encoding="application/x-tex">{\displaystyle n<m<n{\sqrt {3}}}</annotation>

</semantics>

</math><img src="./6cb811cbc8a7fe617dbeab974a6538c497d718cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.125ex; height:2.843ex;" alt="{\displaystyle n<m<n{\sqrt {3}}}" loading="lazy">  (if the quantity inside the <a href="Absolute_value" title="Absolute value">absolute value</a> signs is negative) or  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>(2+{\sqrt {3}})n}">

<msqrt>

</msqrt>

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<annotation encoding="application/x-tex">{\displaystyle m>(2+{\sqrt {3}})n}</annotation>

</semantics>

</math><img src="./c772465339f82670dceacc855db1566eb448d985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.444ex; height:3.009ex;" alt="{\displaystyle m>(2+{\sqrt {3}})n}" loading="lazy"> (if that quantity is positive) to satisfy the <a href="Triangle_inequality" title="Triangle inequality">triangle inequality</a>.

An important characteristic of the automedian triangle is that the squares of its sides form an <a href="Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a>. Specifically, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}-a^{2}=b^{2}-c^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle c^{2}-a^{2}=b^{2}-c^{2}}</annotation>

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</math><img src="./b3e3a1d3cadc7eb947e522a128093df89204fcc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.237ex; height:2.843ex;" alt="{\displaystyle c^{2}-a^{2}=b^{2}-c^{2}}" loading="lazy"> so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2c^{2}=a^{2}+b^{2}.}">

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<annotation encoding="application/x-tex">{\displaystyle 2c^{2}=a^{2}+b^{2}.}</annotation>

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</math><img src="./ec64ceddb0c5e7e47213d3d6609f625a8306d5e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.145ex; height:2.843ex;" alt="{\displaystyle 2c^{2}=a^{2}+b^{2}.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Integer_triangles_with_specific_angle_properties">Integer triangles with specific angle properties</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Integer_triangles_with_a_rational_angle_bisector">Integer triangles with a rational angle bisector</h3></div>

A triangle family with integer sides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation>

</semantics>

</math><img src="./f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"> and with rational bisector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> of angle A is given by<a href="#cite_note-24">[24]</a>

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<annotation encoding="application/x-tex">{\displaystyle a=2(k^{2}-m^{2}),}</annotation>

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</math><img src="./9e7b3461c4b715b08558a493fa697f63f4753988.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.147ex; height:3.176ex;" alt="{\displaystyle a=2(k^{2}-m^{2}),}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle b=(k-m)^{2},}</annotation>

</semantics>

</math><img src="./29fa4d5c5fb8bc6e2ede12fef586d9695a659af2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.698ex; height:3.176ex;" alt="{\displaystyle b=(k-m)^{2},}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle c=(k+m)^{2},}</annotation>

</semantics>

</math><img src="./cd3a49b8deaedcfccbb3263918d476ff96f67d3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.708ex; height:3.176ex;" alt="{\displaystyle c=(k+m)^{2},}" loading="lazy"></dd>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle d={\frac {2km(k^{2}-m^{2})}{k^{2}+m^{2}}},}</annotation>

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</math><img src="./2fc101322973ad2534784717cc7dd50e5d4ee36e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.221ex; height:6.343ex;" alt="{\displaystyle d={\frac {2km(k^{2}-m^{2})}{k^{2}+m^{2}}},}" loading="lazy"></dd></dl>

with integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>m>0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k>m>0}</annotation>

</semantics>

</math><img src="./b5f071081ed64de27bed3e6c2ad7f70583d02b05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.611ex; height:2.176ex;" alt="{\displaystyle k>m>0}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Integer_triangles_with_integer_n-sectors_of_all_angles">Integer triangles with integer n-sectors of all angles</h3></div>

There exist infinitely many non-<a href="Similar_triangles" class="mw-redirect" title="Similar triangles">similar</a> triangles in which the three sides and the bisectors of each of the three angles are integers.<a href="#cite_note-DeBruyn-25">[25]</a>

There exist infinitely many non-similar triangles in which the three sides and the two trisectors of each of the three angles are integers.<a href="#cite_note-DeBruyn-25">[25]</a>

However, for n > 3 there exist no triangles in which the three sides and the (n – 1) n-sectors of each of the three angles are integers.<a href="#cite_note-DeBruyn-25">[25]</a>

<div class="mw-heading mw-heading3"><h3 id="Integer_triangles_with_one_angle_with_a_given_rational_cosine">Integer triangles with one angle with a given rational cosine</h3></div>

Integer triangles with one angle at vertex A having given rational cosine h / k (h < 0 or > 0; k > 0) are given by<a href="#cite_note-26">[26]</a>

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<annotation encoding="application/x-tex">{\displaystyle a=p^{2}-2pqh+q^{2}k^{2},}</annotation>

</semantics>

</math><img src="./0ca415c107b867ff7c30e4886f8555e6f27697c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.019ex; height:3.009ex;" alt="{\displaystyle a=p^{2}-2pqh+q^{2}k^{2},}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=p^{2}-q^{2}k^{2},}</annotation>

</semantics>

</math><img src="./a5e0327807184637d94257e5cbc672ffc73e5637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.206ex; height:3.009ex;" alt="{\displaystyle b=p^{2}-q^{2}k^{2},}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=2qk(p-qh),}</annotation>

</semantics>

</math><img src="./181c4eeacc3620602080eb12cc6f2174c241636f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.423ex; height:2.843ex;" alt="{\displaystyle c=2qk(p-qh),}" loading="lazy"></dd></dl>

where p and q are any coprime positive integers such that p > qk. All primitive solutions can be obtained by dividing a, b, and c by their greatest common divisor.

<div class="mw-heading mw-heading4"><h4 id="Integer_triangles_with_a_60°_angle_(angles_in_arithmetic_progression)">Integer triangles with a 60° angle (angles in arithmetic progression)</h4></div>

All integer triangles with a 60° angle have their angles in an arithmetic progression. All such triangles are proportional to:<a href="#cite_note-Zelator-5">[5]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=4mn,}</annotation>

</semantics>

</math><img src="./ae1b69358e6566dca9f8bc1c01c55b0948862808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.573ex; height:2.509ex;" alt="{\displaystyle a=4mn,}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=3m^{2}+n^{2},}</annotation>

</semantics>

</math><img src="./d57ed6314a483fa819dbdf6524225aff26357b07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.289ex; height:3.009ex;" alt="{\displaystyle b=3m^{2}+n^{2},}" loading="lazy"></dd>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=2mn+|3m^{2}-n^{2}|}</annotation>

</semantics>

</math><img src="./92795d146f77de29f2ed14f0a263ebad412cb5d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.383ex; height:3.176ex;" alt="{\displaystyle c=2mn+|3m^{2}-n^{2}|}" loading="lazy"></dd></dl>

with coprime integers m, n and 1 ≤ n ≤ m or 3m ≤ n. From here, all primitive solutions can be obtained by dividing a, b, and c by their greatest common divisor.

Integer triangles with a 60° angle can also be generated by<a href="#cite_note-27">[27]</a>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=m^{2}-mn+n^{2},}</annotation>

</semantics>

</math><img src="./d2c6ca7abe63694895f2cfd1723880b4dc9d47e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.635ex; height:3.009ex;" alt="{\displaystyle a=m^{2}-mn+n^{2},}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=2mn-n^{2},}</annotation>

</semantics>

</math><img src="./d1bdf38f0a2e3dd5a0889fb483521f55c212f2e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.63ex; height:3.009ex;" alt="{\displaystyle b=2mn-n^{2},}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=m^{2}-n^{2},}</annotation>

</semantics>

</math><img src="./0d0b0fdf7943e4763af17391298eaada8d33b883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.136ex; height:3.009ex;" alt="{\displaystyle c=m^{2}-n^{2},}" loading="lazy"></dd></dl>

with coprime integers m, n with 0 < n < m (the angle of 60° is opposite to the side of length a). From here, all primitive solutions can be obtained by dividing a, b, and c by their greatest common divisor (e.g. an equilateral triangle solution is obtained by taking m = 2 and n = 1, but this produces a = b = c = 3, which is not a primitive solution). See also <a href="#cite_note-Burn-28">[28]</a><a href="#cite_note-Read-29">[29]</a>

More precisely, If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\equiv -n\!{\pmod {3}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\equiv -n\!{\pmod {3}}}</annotation>

</semantics>

</math><img src="./12e7fb5c2c63a2ff6acf5fa3e37227aff81cd45a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.801ex; height:2.843ex;" alt="{\displaystyle m\equiv -n\!{\pmod {3}}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(a,b,c)=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gcd(a,b,c)=3}</annotation>

</semantics>

</math><img src="./c8f12f04cf3d8bce6674adf925ad24b14973f08e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.86ex; height:2.843ex;" alt="{\displaystyle \gcd(a,b,c)=3}" loading="lazy">, otherwise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(a,b,c)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gcd(a,b,c)=1}</annotation>

</semantics>

</math><img src="./ffc7659d40fa17e4084020340c569ea2b15b05ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.86ex; height:2.843ex;" alt="{\displaystyle \gcd(a,b,c)=1}" loading="lazy">. Two different pairs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m,n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m,n)}</annotation>

</semantics>

</math><img src="./274d4857135a7d28a94ba9ee8135779615084d43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle (m,n)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m,m-n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m,m-n)}</annotation>

</semantics>

</math><img src="./c5b2952ba42264809aa8948432c38640eabac453.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.159ex; height:2.843ex;" alt="{\displaystyle (m,m-n)}" loading="lazy"> generate the same triple. Unfortunately the two pairs can both have a gcd of 3, so we can't avoid duplicates by simply skipping that case. Instead, duplicates can be avoided by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> going only till <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m/2}</annotation>

</semantics>

</math><img src="./2d6af4e80f7ef59ed62ebada0b02f1ef4b2a2016.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.365ex; height:2.843ex;" alt="{\displaystyle m/2}" loading="lazy">. We still need to divide by 3 if the gcd is 3. The only solution for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=m/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=m/2}</annotation>

</semantics>

</math><img src="./9e631a07e73613f5b1dd6435bb637a35f96bb362.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.858ex; height:2.843ex;" alt="{\displaystyle n=m/2}" loading="lazy"> under the above constraints is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,3,3)\equiv (1,1,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (3,3,3)\equiv (1,1,1)}</annotation>

</semantics>

</math><img src="./2ff040fc8ff3e1f845e598f701f3a6e4f5dfa0be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.828ex; height:2.843ex;" alt="{\displaystyle (3,3,3)\equiv (1,1,1)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=2,n=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=2,n=1}</annotation>

</semantics>

</math><img src="./6d873bcd3c2c79e247f665791d45f5b6da979bac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.991ex; height:2.509ex;" alt="{\displaystyle m=2,n=1}" loading="lazy">. With this additional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq m/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\leq m/2}</annotation>

</semantics>

</math><img src="./d14b7aef1aa7c3f901b9519627e3df52821b6aa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.858ex; height:2.843ex;" alt="{\displaystyle n\leq m/2}" loading="lazy"> constraint all triples can be generated uniquely.

An <a href="Eisenstein_triple" title="Eisenstein triple">Eisenstein triple</a> is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 degrees.

<div class="mw-heading mw-heading4"><h4 id="Integer_triangles_with_a_120°_angle">Integer triangles with a 120° angle</h4></div>

Integer triangles with a 120° angle can be generated by<a href="#cite_note-30">[30]</a>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=m^{2}+mn+n^{2},}</annotation>

</semantics>

</math><img src="./6459545347543de76882201eaa443065e60f5362.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.635ex; height:3.009ex;" alt="{\displaystyle a=m^{2}+mn+n^{2},}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=2mn+n^{2},}</annotation>

</semantics>

</math><img src="./22c1eea75b008bd127619a83104933a5900a38a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.63ex; height:3.009ex;" alt="{\displaystyle b=2mn+n^{2},}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=m^{2}-n^{2},}</annotation>

</semantics>

</math><img src="./0d0b0fdf7943e4763af17391298eaada8d33b883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.136ex; height:3.009ex;" alt="{\displaystyle c=m^{2}-n^{2},}" loading="lazy"></dd></dl>

with coprime integers m, n with 0 < n < m (the angle of 120° is opposite to the side of length a). From here, all primitive solutions can be obtained by dividing a, b, and c by their greatest common divisor. The smallest solution, for m = 2 and n = 1, is the triangle with sides (3,5,7). See also.<a href="#cite_note-Burn-28">[28]</a><a href="#cite_note-Read-29">[29]</a>

More precisely, If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\equiv n\!{\pmod {3}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\equiv n\!{\pmod {3}}}</annotation>

</semantics>

</math><img src="./2c24166cfea30f67972a13e5159cabab7f24c03c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.993ex; height:2.843ex;" alt="{\displaystyle m\equiv n\!{\pmod {3}}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(a,b,c)=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gcd(a,b,c)=3}</annotation>

</semantics>

</math><img src="./c8f12f04cf3d8bce6674adf925ad24b14973f08e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.86ex; height:2.843ex;" alt="{\displaystyle \gcd(a,b,c)=3}" loading="lazy">, otherwise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(a,b,c)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gcd(a,b,c)=1}</annotation>

</semantics>

</math><img src="./ffc7659d40fa17e4084020340c569ea2b15b05ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.86ex; height:2.843ex;" alt="{\displaystyle \gcd(a,b,c)=1}" loading="lazy">. Since the biggest side a can only be generated with a single <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m,n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m,n)}</annotation>

</semantics>

</math><img src="./274d4857135a7d28a94ba9ee8135779615084d43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle (m,n)}" loading="lazy"> pair, each primitive triple can be generated in precisely two ways: once directly with a gcd of 1, and once indirectly with a gcd of 3. Therefore, in order to generate all primitive triples uniquely, one can just add additional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\not \equiv n\!{\pmod {3}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\not \equiv n\!{\pmod {3}}}</annotation>

</semantics>

</math><img src="./c6b19fa14a8173fcb2723dcc4d6948e56dd64b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.994ex; height:2.843ex;" alt="{\displaystyle m\not \equiv n\!{\pmod {3}}}" loading="lazy"> condition.

<div class="mw-heading mw-heading3"><h3 id="Integer_triangles_with_one_angle_equal_to_an_arbitrary_rational_number_times_another_angle">Integer triangles with one angle equal to an arbitrary rational number times another angle</h3></div>

For positive coprime integers h and k, the triangle with the following sides has angles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\alpha }</annotation>

</semantics>

</math><img src="./5713ef8643b080aa7087a97cccfa2b3754b7c965.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.827ex; height:2.176ex;" alt="{\displaystyle h\alpha }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\alpha }</annotation>

</semantics>

</math><img src="./ea9f16961e319eed62e2b4a79ea9830e740e7a44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.699ex; height:2.176ex;" alt="{\displaystyle k\alpha }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi -(h+k)\alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi -(h+k)\alpha }</annotation>

</semantics>

</math><img src="./2e8dacb5c03d1f9311dcc951e1f7705d698b12c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.86ex; height:2.843ex;" alt="{\displaystyle \pi -(h+k)\alpha }" loading="lazy"> and hence two angles in the ratio h : k, and its sides are integers:<a href="#cite_note-31">[31]</a>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=q^{h+k-1}{\frac {\sin h\alpha }{\sin \alpha }}=q^{k}\cdot \sum _{0\leq i\leq {\frac {h-1}{2}}}(-1)^{i}{\binom {h}{2i+1}}p^{h-2i-1}(q^{2}-p^{2})^{i},}</annotation>

</semantics>

</math><img src="./d9d398af0ce81491a3fd17aab49aae5d9fd8023d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:64.469ex; height:8.343ex;" alt="{\displaystyle a=q^{h+k-1}{\frac {\sin h\alpha }{\sin \alpha }}=q^{k}\cdot \sum _{0\leq i\leq {\frac {h-1}{2}}}(-1)^{i}{\binom {h}{2i+1}}p^{h-2i-1}(q^{2}-p^{2})^{i},}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<mrow>

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<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=q^{h+k-1}{\frac {\sin k\alpha }{\sin \alpha }}=q^{h}\cdot \sum _{0\leq i\leq {\frac {k-1}{2}}}(-1)^{i}{\binom {k}{2i+1}}p^{k-2i-1}(q^{2}-p^{2})^{i},}</annotation>

</semantics>

</math><img src="./60328662f8021e12b1ca313a06ab25a91ab004ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:64.036ex; height:8.343ex;" alt="{\displaystyle b=q^{h+k-1}{\frac {\sin k\alpha }{\sin \alpha }}=q^{h}\cdot \sum _{0\leq i\leq {\frac {k-1}{2}}}(-1)^{i}{\binom {k}{2i+1}}p^{k-2i-1}(q^{2}-p^{2})^{i},}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=q^{h+k-1}{\frac {\sin(h+k)\alpha }{\sin \alpha }}=\sum _{0\leq i\leq {\frac {h+k-1}{2}}}(-1)^{i}{\binom {h+k}{2i+1}}p^{h+k-2i-1}(q^{2}-p^{2})^{i},}</annotation>

</semantics>

</math><img src="./77a6ae0251914052f75766ab00fbc0539cd5ad31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:69.741ex; height:8.509ex;" alt="{\displaystyle c=q^{h+k-1}{\frac {\sin(h+k)\alpha }{\sin \alpha }}=\sum _{0\leq i\leq {\frac {h+k-1}{2}}}(-1)^{i}{\binom {h+k}{2i+1}}p^{h+k-2i-1}(q^{2}-p^{2})^{i},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\cos ^{-1}\!{\frac {p}{q}}}">

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =\cos ^{-1}\!{\frac {p}{q}}}</annotation>

</semantics>

</math><img src="./ff9c15bf8f80b1d86c9e2e2f2ec7b705891224c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.036ex; height:5.343ex;" alt="{\displaystyle \alpha =\cos ^{-1}\!{\frac {p}{q}}}" loading="lazy"> and p and q are any coprime integers such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\pi }{h+k}}<{\frac {p}{q}}<1}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\pi }{h+k}}<{\frac {p}{q}}<1}</annotation>

</semantics>

</math><img src="./e1bb648bf7571a3c0b6b457ce90312bd715dcc25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.09ex; height:5.343ex;" alt="{\displaystyle \cos {\frac {\pi }{h+k}}<{\frac {p}{q}}<1}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Integer_triangles_with_one_angle_equal_to_twice_another">Integer triangles with one angle equal to twice another</h4></div>

With angle A opposite side <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and angle B opposite side <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">, some triangles with B = 2A are generated by<a href="#cite_note-Deshpande-32">[32]</a>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=n^{2},}</annotation>

</semantics>

</math><img src="./b7282f6c55e60cf38e0bf3e34b720d3c54768997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.424ex; height:3.009ex;" alt="{\displaystyle a=n^{2},}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=mn,}</annotation>

</semantics>

</math><img src="./2898cd6b25d95e13f940e4e6d36994d40218e081.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.178ex; height:2.509ex;" alt="{\displaystyle b=mn,}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=m^{2}-n^{2},}</annotation>

</semantics>

</math><img src="./0d0b0fdf7943e4763af17391298eaada8d33b883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.136ex; height:3.009ex;" alt="{\displaystyle c=m^{2}-n^{2},}" loading="lazy"></dd></dl>

with integers m, n such that 0 < n < m < 2n.

All triangles with B = 2A (whether integer or not) satisfy<a href="#cite_note-33">[33]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(a+c)=b^{2}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a(a+c)=b^{2}.}</annotation>

</semantics>

</math><img src="./3da5fe08a338f7542aa77d3b9047572892dc601e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.913ex; height:3.176ex;" alt="{\displaystyle a(a+c)=b^{2}.}" loading="lazy">

<div class="mw-heading mw-heading4"><h4 id="Integer_triangles_with_one_angle_equal_to_3/2_times_another">Integer triangles with one angle equal to 3/2 times another</h4></div>

The <a href="Equivalence_class" title="Equivalence class">equivalence class</a> of similar triangles with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\tfrac {3}{2}}A}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B={\tfrac {3}{2}}A}</annotation>

</semantics>

</math><img src="./87d6ebe37b42aebf97ad203b8774bb9ef23bb333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.264ex; height:3.509ex;" alt="{\displaystyle B={\tfrac {3}{2}}A}" loading="lazy"> are generated by<a href="#cite_note-Deshpande-32">[32]</a>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=mn^{3},}</annotation>

</semantics>

</math><img src="./82410667e381c67da21a7662587e44f21c79740f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.464ex; height:3.009ex;" alt="{\displaystyle a=mn^{3},}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=n^{2}(m^{2}-n^{2}),}</annotation>

</semantics>

</math><img src="./b973b8266f3003abd72ec944cff0a197b94d0333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.385ex; height:3.176ex;" alt="{\displaystyle b=n^{2}(m^{2}-n^{2}),}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=(m^{2}-n^{2})^{2}-m^{2}n^{2},}</annotation>

</semantics>

</math><img src="./d54ed528f76fe61cfba0bace515dc74369510b1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.384ex; height:3.176ex;" alt="{\displaystyle c=(m^{2}-n^{2})^{2}-m^{2}n^{2},}" loading="lazy"></dd></dl>

with integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,n}</annotation>

</semantics>

</math><img src="./6568e95b6bf8f39b7fd2c9b52b7b00ee124c6250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.469ex; height:2.009ex;" alt="{\displaystyle m,n}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\varphi n<m<2n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<\varphi n<m<2n}</annotation>

</semantics>

</math><img src="./97ce50c024d66dc8e62de373cccdb5573de39c38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.97ex; height:2.676ex;" alt="{\displaystyle 0<\varphi n<m<2n}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is the <a href="Golden_ratio" title="Golden ratio">golden ratio</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}{\bigr )}\approx 1.61803}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}{\bigr )}\approx 1.61803}</annotation>

</semantics>

</math><img src="./a9896ab668462f3e721b738bc4b9e67a8cc85cac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.228ex; height:3.509ex;" alt="{\textstyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}{\bigr )}\approx 1.61803}" loading="lazy">.

All triangles with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\tfrac {3}{2}}A}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B={\tfrac {3}{2}}A}</annotation>

</semantics>

</math><img src="./87d6ebe37b42aebf97ad203b8774bb9ef23bb333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.264ex; height:3.509ex;" alt="{\displaystyle B={\tfrac {3}{2}}A}" loading="lazy"> (whether with integer sides or not) satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b^{2}-a^{2})(b^{2}-a^{2}+bc)=a^{2}c^{2}.}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (b^{2}-a^{2})(b^{2}-a^{2}+bc)=a^{2}c^{2}.}</annotation>

</semantics>

</math><img src="./ce14a0775d279c4de3177b07f133b6e6b473dfad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.906ex; height:3.176ex;" alt="{\displaystyle (b^{2}-a^{2})(b^{2}-a^{2}+bc)=a^{2}c^{2}.}" loading="lazy">

<div class="mw-heading mw-heading4"><h4 id="Integer_triangles_with_one_angle_three_times_another">Integer triangles with one angle three times another</h4></div>

We can generate the full equivalence class of similar triangles that satisfy B = 3A by using the formulas<a href="#cite_note-34">[34]</a>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=n^{3},\,}</annotation>

</semantics>

</math><img src="./f37d51ce06e45eb6a43e11d0410381fa849e324b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.811ex; height:3.009ex;" alt="{\displaystyle a=n^{3},\,}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=n(m^{2}-n^{2}),\,}</annotation>

</semantics>

</math><img src="./a1b6bab55488f7beaf6a05fea65e3b413cd9e39f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.718ex; height:3.176ex;" alt="{\displaystyle b=n(m^{2}-n^{2}),\,}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=m(m^{2}-2n^{2}),\,}</annotation>

</semantics>

</math><img src="./3965238f3bf33b6b82c35e39c8624f704c010150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.535ex; height:3.176ex;" alt="{\displaystyle c=m(m^{2}-2n^{2}),\,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> are integers such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}n<m<2n}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}n<m<2n}</annotation>

</semantics>

</math><img src="./8b7cbadf582790f9d2644343235fe5ed674aa38a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.288ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}n<m<2n}" loading="lazy">.

All triangles with B = 3A (whether with integer sides or not) satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ac^{2}=(b-a)^{2}(b+a).}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ac^{2}=(b-a)^{2}(b+a).}</annotation>

</semantics>

</math><img src="./37533a3cbfdd67f5b353b1991062d4cf694e10e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.845ex; height:3.176ex;" alt="{\displaystyle ac^{2}=(b-a)^{2}(b+a).}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Integer_triangles_with_three_rational_angles">Integer triangles with three rational angles</h3></div>

The only integer triangle with three rational angles (rational numbers of degrees, or equivalently rational fractions of a full turn) is the <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>.<a href="#cite_note-CG-2">[2]</a> This is because integer sides imply three rational <a href="Cosine" class="mw-redirect" title="Cosine">cosines</a> by the <a href="Law_of_cosines" title="Law of cosines">law of cosines</a>, and by <a href="Niven's_theorem" title="Niven's theorem">Niven's theorem</a> a rational cosine coincides with a rational angle if and only if the cosine equals 0, ±1/2, or ±1. The only ones of these giving an angle strictly between 0° and 180° are the cosine value 1/2 with the angle 60°, the cosine value –1/2 with the angle 120°, and the cosine value 0 with the angle 90°. The only combination of three of these, allowing multiple use of any of them and summing to 180°, is three 60° angles.

<div class="mw-heading mw-heading2"><h2 id="Integer_triangles_with_integer_ratio_of_circumradius_to_inradius">Integer triangles with integer ratio of circumradius to inradius</h2></div>

Conditions are known in terms of <a href="Elliptic_curve" title="Elliptic curve">elliptic curves</a> for an integer triangle to have an integer ratio N of the <a href="Circumradius" class="mw-redirect" title="Circumradius">circumradius</a> to the <a href="Inradius" class="mw-redirect" title="Inradius">inradius</a>.<a href="#cite_note-35">[35]</a><a href="#cite_note-36">[36]</a> The smallest case, that of the <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>, has N = 2. In every known case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\equiv 2\!{\pmod {8}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\equiv 2\!{\pmod {8}}}</annotation>

</semantics>

</math><img src="./9eb8fc98dbad634388b6359a18962b8c9f263cfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.784ex; height:2.843ex;" alt="{\displaystyle N\equiv 2\!{\pmod {8}}}" loading="lazy"> – that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N-2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N-2}</annotation>

</semantics>

</math><img src="./9cdfb930783ca4ca0062df54535c6e35d555dd0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N-2}" loading="lazy"> is divisible by 8.

<div class="mw-heading mw-heading2"><h2 id="5-Con_triangle_pairs">5-Con triangle pairs</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="5-Con_triangles" title="5-Con triangles">5-Con triangles</a></div>

A 5-Con triangle pair is a pair of triangles that are <a href="Similarity_(geometry)" title="Similarity (geometry)">similar</a> but not <a href="Congruence_(geometry)" title="Congruence (geometry)">congruent</a> and that share three angles and two sidelengths. Primitive integer 5-Con triangles, in which the four distinct integer sides (two sides each appearing in both triangles, and one other side in each triangle) share no prime factor, have triples of sides

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x^{3},x^{2}y,xy^{2})}</annotation>

</semantics>

</math><img src="./8fd5f0372d8c0c93b34b4f9af9e6f772f0cdefaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.345ex; height:3.176ex;" alt="{\displaystyle (x^{3},x^{2}y,xy^{2})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{2}y,xy^{2},y^{3})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x^{2}y,xy^{2},y^{3})}</annotation>

</semantics>

</math><img src="./8860c38c9c6def0f18bca38ed8beeea9b888e9cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.176ex; height:3.176ex;" alt="{\displaystyle (x^{2}y,xy^{2},y^{3})}" loading="lazy"></dd></dl>

for positive coprime integers x and y. The smallest example is the pair (8, 12, 18), (12, 18, 27), generated by x = 2, y = 3.

<div class="mw-heading mw-heading2"><h2 id="Particular_integer_triangles">Particular integer triangles</h2></div>

<ul><li>The only triangle with consecutive integers for sides and area has sides (3, 4, 5) and area 6.</li>

<li>The only triangle with consecutive integers for an altitude and the sides has sides (13, 14, 15) and altitude from side 14 equal to 12.</li>

<li>The (2, 3, 4) triangle and its multiples are the only triangles with integer sides in arithmetic progression and having the complementary exterior angle property.<a href="#cite_note-37">[37]</a><a href="#cite_note-38">[38]</a><a href="#cite_note-Mitchell_2:3:4-39">[39]</a> This property states that if angle C is obtuse and if a segment is dropped from B meeting perpendicularly AC <a href="Extended_side" title="Extended side">extended</a> at P, then ∠CAB=2∠CBP.</li>

<li>The (3, 4, 5) triangle and its multiples are the only integer right triangles having sides in arithmetic progression.<a href="#cite_note-Mitchell_2:3:4-39">[39]</a></li>

<li>The (4, 5, 6) triangle and its multiples are the only triangles with one angle being twice another and having integer sides in arithmetic progression.<a href="#cite_note-Mitchell_2:3:4-39">[39]</a></li>

<li>The (3, 5, 7) triangle and its multiples are the only triangles with a 120° angle and having integer sides in arithmetic progression.<a href="#cite_note-Mitchell_2:3:4-39">[39]</a></li>

<li>The only integer triangle with area = semiperimeter<a href="#cite_note-MacHale,_D._1989-40">[40]</a> has sides (3, 4, 5).</li>

<li>The only integer triangles with area = perimeter have sides<a href="#cite_note-MacHale,_D._1989-40">[40]</a><a href="#cite_note-41">[41]</a> (5, 12, 13), (6, 8, 10), (6, 25, 29), (7, 15, 20), and (9, 10, 17). Of these the first two, but not the last three, are right triangles.</li>

<li>There exist integer triangles with three rational <a href="Median_(geometry)" title="Median (geometry)">medians</a>.<a href="#cite_note-Sierpinski-9">[9]</a>: p. 64 The smallest has sides (68, 85, 87). Others include (127, 131, 158), (113, 243, 290), (145, 207, 328) and (327, 386, 409).</li>

<li>There are no isosceles Pythagorean triangles.<a href="#cite_note-Sastry-15">[15]</a></li>

<li>The only primitive Pythagorean triangles for which the square of the perimeter equals an integer multiple of the area are (3, 4, 5) with perimeter 12 and area 6 and with the ratio of perimeter squared to area being 24; (5, 12, 13) with perimeter 30 and area 30 and with the ratio of perimeter squared to area being 30; and (9, 40, 41) with perimeter 90 and area 180 and with the ratio of perimeter squared to area being 45.<a href="#cite_note-42">[42]</a></li>

<li>There exists a unique (up to similitude) pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. The unique pair consists of the (377, 135, 352) triangle and the (366, 366, 132) triangle.<a href="#cite_note-:0-43">[43]</a> There is no pair of such triangles if the triangles are also required to be primitive integral triangles.<a href="#cite_note-:0-43">[43]</a> The authors stress the striking fact that the second assertion can be proved by an elementary argumentation (they do so in their appendix A), whilst the first assertion needs modern highly non-trivial mathematics.</li></ul>

<ul><li><a href="Brahmagupta_triangle" title="Brahmagupta triangle">Brahmagupta triangle</a>, a <a href="Heronian_triangle" title="Heronian triangle">Heronian triangle</a> in which the side lengths are consecutive integers</li>

<li><a href="Robbins_pentagon" title="Robbins pentagon">Robbins pentagon</a>, a cyclic pentagon with integer sides and integer area</li>

<li><a href="Euler_brick" title="Euler brick">Euler brick</a>, a cuboid with integer edges and integer face diagonals</li>

<li><a href="Tetrahedron#Integer_tetrahedra" title="Tetrahedron">Tetrahedron § Integer tetrahedra</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div id="Geometry874" style="font-size:114%;margin:0 4em"><a href="Geometry" title="Geometry">Geometry</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_geometry" title="History of geometry">History</a>

<ul><li><a href="Timeline_of_geometry" title="Timeline of geometry">Timeline</a></li></ul></li>

<li><a href="Lists_of_geometry_topics" class="mw-redirect" title="Lists of geometry topics">Lists of geometry topics</a></li>

<li><a href="Foundations_of_geometry" title="Foundations of geometry">Foundations of geometry</a></li>

<li><a href="Outline_of_geometry" title="Outline of geometry">Outline of geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_geometry" title="Convex geometry">Convex</a></li>

<li><a href="Discrete_geometry" title="Discrete geometry">Discrete</a></li>

<li><a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane geometry</a></li>

<li><a href="Solid_geometry" title="Solid geometry">Solid</a></li>

<li><a href="Affine_geometry" title="Affine geometry">Affine</a></li>

<li><a href="Trigonometry" title="Trigonometry">Trigonometry</a>

<ul><li><a href="Spherical_trigonometry" title="Spherical trigonometry">Spherical</a></li>

<li><a href="Generalized_trigonometry" title="Generalized trigonometry">Generalized</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamental Concepts (Euclidean)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Point_(geometry)" title="Point (geometry)">Point</a></li>

<li><a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li>

<li><a href="Space_(mathematics)" title="Space (mathematics)">Space</a></li>

<li><a href="Distance" title="Distance">Distance</a></li>

<li><a href="Angle" title="Angle">Angle</a></li>

<li><a href="Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li>

<li><a href="Perpendicular" title="Perpendicular">Perpendicular</a></li>

<li><a href="Triangle" title="Triangle">Triangle</a></li>

<li><a href="Circle" title="Circle">Circle</a></li>

<li><a href="Polygon" title="Polygon">Polygon</a></li>

<li><a href="Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li>

<li><a href="Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li>

<li><a href="Euclid's_Elements" title="Euclid's Elements">Euclid's Elements</a></li>

<li><a href="Euclidean_space" title="Euclidean space">Euclidean space</a></li>

<li><a href="Coordinate_system" title="Coordinate system">Coordinate system</a>

<ul><li><a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian</a></li>

<li><a href="Polar_coordinate_system" title="Polar coordinate system">Polar</a></li></ul></li>

<li><a href="Dimension_(mathematics_and_physics)" class="mw-redirect" title="Dimension (mathematics and physics)">Dimension</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Non-Euclidean_space" class="mw-redirect" title="Non-Euclidean space">Non-Euclidean space</a></li>

<li><a href="Elliptic_geometry" title="Elliptic geometry">Elliptic</a></li>

<li><a href="Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li>

<li><a href="Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Based on Methods or Structures</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebraic_geometry" title="Algebraic geometry">Algebraic</a>

<ul><li><a href="Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li>

<li><a href="Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential</a>

<ul><li><a href="Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li>

<li><a href="Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li></ul></li>

<li><a href="Absolute_geometry" title="Absolute geometry">Absolute</a></li>

<li><a href="Analytic_geometry" title="Analytic geometry">Analytic geometry</a></li>

<li><a href="Complex_geometry" title="Complex geometry">Complex</a></li>

<li><a href="Computational_geometry" title="Computational geometry">Computational</a></li>

<li><a href="Conformal_geometry" title="Conformal geometry">Conformal</a></li>

<li><a href="Discrete_geometry" title="Discrete geometry">Discrete</a></li>

<li><a href="Fractal" title="Fractal">Fractal</a></li>

<li><a href="Geometric_group_theory" title="Geometric group theory">Geometric group theory</a></li>

<li><a href="Information_geometry" title="Information geometry">Information</a></li>

<li><a href="Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean</a></li>

<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative</a></li>

<li><a href="Projective_geometry" title="Projective geometry">Projective</a></li>

<li><a href="Spectral_geometry" title="Spectral geometry">Spectral</a></li>

<li><a href="Thurston_geometry" class="mw-redirect" title="Thurston geometry">Thurston</a></li>

<li><a href="Metric_geometry" class="mw-redirect" title="Metric geometry">Metric geometry</a></li>

<li><a href="Geometric_measure_theory" title="Geometric measure theory">Geometric measure theory</a></li>

<li><a href="Geometric_analysis" title="Geometric analysis">Geometric analysis</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topology</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Topology" title="Topology">Topology</a>

<ul><li><a href="General_topology" title="General topology">General</a></li>

<li><a href="Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic</a></li>

<li><a href="Continuum_(topology)" title="Continuum (topology)">Continuum</a></li></ul></li>

<li><a href="Algebraic_topology" title="Algebraic topology">Algebraic</a>

<ul><li><a href="Noncommutative_topology" title="Noncommutative topology">Noncommutative</a></li></ul></li>

<li><a href="Differential_topology" title="Differential topology">Differential</a>

<ul><li><a href="Geometric_topology" title="Geometric topology">Geometric</a></li>

<li><a href="Low-dimensional_topology" title="Low-dimensional topology">Low-dimensional</a></li></ul></li>

<li><a href="Combinatorial_topology" title="Combinatorial topology">Combinatorial</a></li>

<li><a href="Topological_dynamics" title="Topological dynamics">Topological dynamics</a></li>

<li><a href="Knot_theory" title="Knot theory">Knot Theory</a></li>

<li><a href="Symplectic_topology" class="mw-redirect" title="Symplectic topology">Symplectic Topology</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lists_of_geometry_topics" class="mw-redirect" title="Lists of geometry topics">Lists of geometry topics</a></li>

<li><a href="List_of_combinatorial_computational_geometry_topics" title="List of combinatorial computational geometry topics">Combinatorial computational geometry topics</a></li>

<li><a href="List_of_differential_geometry_topics" title="List of differential geometry topics">Differential geometry topics</a></li>

<li><a href="List_of_formulas_in_elementary_geometry" title="List of formulas in elementary geometry">Formulas in elementary geometry</a></li>

<li><a href="List_of_formulas_in_Riemannian_geometry" title="List of formulas in Riemannian geometry">Formulas in Riemannian geometry</a></li>

<li><a href="List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">Geometry topics</a></li>

<li><a href="List_of_knot_theory_topics" title="List of knot theory topics">Knot theory topics</a></li>

<li><a href="List_of_numerical_computational_geometry_topics" title="List of numerical computational geometry topics">Numerical computational geometry topics</a></li>

<li><a href="List_of_polygons" title="List of polygons">Polygons</a></li>

<li><a href="List_of_shapes" class="mw-redirect" title="List of shapes">Shapes</a></li>

<li><a href="List_of_topologies" title="List of topologies">Topologies</a></li>

<li><a href="List_of_examples_in_general_topology" title="List of examples in general topology">General topology</a></li>

<li><a href="Outline_of_geometry" title="Outline of geometry">Outline of geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Glossaries</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Glossary_of_algebraic_geometry" title="Glossary of algebraic geometry">Algebraic geometry</a></li>

<li><a href="Glossary_of_algebraic_topology" title="Glossary of algebraic topology">Algebraic topology</a></li>

<li><a href="Glossary_of_arithmetic_and_diophantine_geometry" title="Glossary of arithmetic and diophantine geometry">Arithmetic and diophantine geometry</a></li>

<li><a href="Glossary_of_classical_algebraic_geometry" title="Glossary of classical algebraic geometry">Classical algebraic geometry</a></li>

<li><a href="Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Differential geometry and topology</a></li>

<li><a href="Glossary_of_general_topology" title="Glossary of general topology">Topology</a></li>

<li><a href="Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Mathematical symbols</a></li>

<li><a href="Glossary_of_Riemannian_and_metric_geometry" title="Glossary of Riemannian and metric geometry">Riemannian and metric geometry</a></li>

<li><a href="Glossary_of_shapes_with_metaphorical_names" title="Glossary of shapes with metaphorical names">Shapes with metaphorical names</a></li>

<li><a href="Glossary_of_symplectic_geometry" title="Glossary of symplectic geometry">Symplectic geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Finite_geometry" title="Finite geometry">Finite geometry</a></li>

<li><a href="Incidence_geometry" title="Incidence geometry">Incidence geometry</a></li>

<li><a href="Ordered_geometry" title="Ordered geometry">Ordered geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li>

<li><a href="Group_theory" title="Group theory">Group theory</a></li>

<li><a href="Ring_theory" title="Ring theory">Ring theory</a></li>

<li><a href="Field_(mathematics)" title="Field (mathematics)">Field theory</a></li>

<li><a href="Calculus_on_Euclidean_space#Calculus_on_manifolds" title="Calculus on Euclidean space">Calculus on manifolds</a></li>

<li><a href="Vector_calculus" title="Vector calculus">Vector calculus</a></li>

<li><a href="Fractal_analysis" title="Fractal analysis">Fractal analysis</a></li>

<li><a href="Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li>

<li><a href="Geometry_of_numbers" title="Geometry of numbers">Geometry of numbers</a></li>

<li><a href="Lattice_theory" class="mw-redirect" title="Lattice theory">Lattice theory</a></li>

<li><a href="Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li>

<li><a href="Statistical_shape_analysis" title="Statistical shape analysis">Statistical shape analysis</a></li>

<li><a href="Spatial_statistics" title="Spatial statistics">Spatial statistics</a></li>

<li><a href="Geometric_data_analysis" title="Geometric data analysis">Geometric data analysis</a></li>

<li><a href="Foundations_of_geometry" title="Foundations of geometry">Foundations of geometry</a></li>

<li><a href="Axiomatic_system" title="Axiomatic system">Axiomatic system</a></li>

<li><a href="Constructive_geometry" class="mw-redirect" title="Constructive geometry">Constructive geometry</a></li>

<li><a href="Anabelian_geometry" title="Anabelian geometry">Anabelian geometry</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Categories:

<ul><li> Geometry</li>

<li> Trigonometry</li>

<li> Topology</li>

<li> Category:History of geometry</li>

</ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-23" href="https://en.wikipedia.org/wiki/?title=Integer_triangle&oldid=1302154622">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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