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<title>Inscribed angle</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Inscribed angle</span></span>

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<p>In <a href="Geometry" title="Geometry">geometry</a>, an <b>inscribed angle</b> is the <a href="Angle" title="Angle">angle</a> formed in the interior of a <a href="Circle" title="Circle">circle</a> when two <a href="Chord_(geometry)" title="Chord (geometry)">chords</a> intersect on the circle. It can also be defined as the angle <a href="Subtend" class="mw-redirect" title="Subtend">subtended</a> at a point on the circle by two given points on the circle.

</p><p>Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint.

</p><p>The <b>inscribed angle theorem</b> relates the <a href="Angle#Measuring_angles" title="Angle">measure</a> of an inscribed angle to that of the <a href="Central_angle" title="Central angle">central angle</a> intercepting the same <a href="Circular_arc" title="Circular arc">arc</a>.

</p><p>The inscribed angle theorem appears as Proposition 20 in Book 3 of <a href="Euclid's_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>.

</p><p>Note that this theorem is not to be confused with the <a href="Angle_bisector_theorem" title="Angle bisector theorem">Angle bisector theorem</a>, which also involves angle bisection (but of an angle of a triangle not inscribed in a circle).

</p>

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

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

<p>The inscribed angle theorem states that an angle <span class="texhtml mvar" style="font-style:italic;">θ</span> inscribed in a circle is half of the central angle <span class="texhtml">2<i>θ</i></span> that <a href="Intercepted_arc" class="mw-redirect" title="Intercepted arc">intercepts</a> the same <a href="Arc_(geometry)" class="mw-redirect" title="Arc (geometry)">arc</a> on the circle. Therefore, the angle does not change as its <a href="Vertex_(geometry)" title="Vertex (geometry)">vertex</a> is moved to different positions on the same arc of the circle.

</p>

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

<div class="mw-heading mw-heading4"><h4 id="Inscribed_angles_where_one_chord_is_a_diameter">Inscribed angles where one chord is a diameter</h4></div>

<p>Let <span class="texhtml mvar" style="font-style:italic;">O</span> be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them <span class="texhtml mvar" style="font-style:italic;">V</span> and <span class="texhtml mvar" style="font-style:italic;">A</span>. Designate point <span class="texhtml mvar" style="font-style:italic;">B</span> to be <a href="Diametrically_opposite" class="mw-redirect" title="Diametrically opposite">diametrically opposite</a> point <span class="texhtml mvar" style="font-style:italic;">V</span>. Draw chord <span class="texhtml mvar" style="font-style:italic;">VB</span>, a diameter containing point <span class="texhtml mvar" style="font-style:italic;">O</span>. Draw chord <span class="texhtml mvar" style="font-style:italic;">VA</span>. Angle <span class="texhtml">∠<i>BVA</i></span> is an inscribed angle that intercepts arc <span class="texhtml mvar" style="font-style:italic;"><span style="line-height: 1.2em; padding-top: 0.2em; border: 1px solid transparent; border-top-color: var(--color-base,#202122); border-top-left-radius: 50% 25%; border-top-right-radius: 50% 25%;">AB</span></span>; denote it as <span class="texhtml mvar" style="font-style:italic;">ψ</span>. Draw line <span class="texhtml mvar" style="font-style:italic;">OA</span>. Angle <span class="texhtml">∠<i>BOA</i></span> is a <a href="Central_angle" title="Central angle">central angle</a> that also intercepts arc <span class="texhtml mvar" style="font-style:italic;"><span style="line-height: 1.2em; padding-top: 0.2em; border: 1px solid transparent; border-top-color: var(--color-base,#202122); border-top-left-radius: 50% 25%; border-top-right-radius: 50% 25%;">AB</span></span>; denote it as <span class="texhtml mvar" style="font-style:italic;">θ</span>.

</p><p>Lines <span class="texhtml mvar" style="font-style:italic;">OV</span> and <span class="texhtml mvar" style="font-style:italic;">OA</span> are both <a href="Radius" title="Radius">radii</a> of the circle, so they have equal lengths. Therefore, triangle <span class="texhtml">△<i>VOA</i></span> is <a href="Isosceles" class="mw-redirect" title="Isosceles">isosceles</a>, so angle <span class="texhtml">∠<i>BVA</i></span> and angle <span class="texhtml">∠<i>VAO</i></span> are equal.

</p><p>Angles <span class="texhtml">∠<i>BOA</i></span> and <span class="texhtml">∠<i>AOV</i></span> are <a href="Supplementary_angle" class="mw-redirect" title="Supplementary angle">supplementary</a>, summing to a <a href="Straight_angle" class="mw-redirect" title="Straight angle">straight angle</a> (180°), so angle <span class="texhtml">∠<i>AOV</i></span> measures <span class="texhtml">180° − <i>θ</i></span>.

</p><p>The three angles of triangle <span class="texhtml">△<i>VOA</i></span> <a href="Sum_of_angles_of_a_triangle" title="Sum of angles of a triangle">must sum to <span class="texhtml">180°</span></a>:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (180^{\circ }-\theta )+\psi +\psi =180^{\circ }.}">

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<annotation encoding="application/x-tex">{\displaystyle (180^{\circ }-\theta )+\psi +\psi =180^{\circ }.}</annotation>

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</p><p>Adding <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta -180^{\circ }}">

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<annotation encoding="application/x-tex">{\displaystyle \theta -180^{\circ }}</annotation>

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</math></span><img src="./07a765acc55329deaf4cfd3c8bafb3f7362144ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.472ex; height:2.509ex;" alt="{\displaystyle \theta -180^{\circ }}" loading="lazy"></span> to both sides yields

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\psi =\theta .}">

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

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<div class="mw-heading mw-heading4"><h4 id="Inscribed_angles_with_the_center_of_the_circle_in_their_interior">Inscribed angles with the center of the circle in their interior</h4></div>

<p>Given a circle whose center is point <span class="texhtml mvar" style="font-style:italic;">O</span>, choose three points <span class="texhtml mvar" style="font-style:italic;">V, C, D</span> on the circle. Draw lines <span class="texhtml mvar" style="font-style:italic;">VC</span> and <span class="texhtml mvar" style="font-style:italic;">VD</span>: angle <span class="texhtml">∠<i>DVC</i></span> is an inscribed angle. Now draw line <span class="texhtml mvar" style="font-style:italic;">OV</span> and extend it past point <span class="texhtml mvar" style="font-style:italic;">O</span> so that it intersects the circle at point <span class="texhtml mvar" style="font-style:italic;">E</span>. Angle <span class="texhtml">∠<i>DVC</i></span> intercepts arc <span class="texhtml mvar" style="font-style:italic;"><span style="line-height: 1.2em; padding-top: 0.2em; border: 1px solid transparent; border-top-color: var(--color-base,#202122); border-top-left-radius: 50% 25%; border-top-right-radius: 50% 25%;">DC</span></span> on the circle.

</p><p>Suppose this arc includes point <span class="texhtml mvar" style="font-style:italic;">E</span> within it. Point <span class="texhtml mvar" style="font-style:italic;">E</span> is diametrically opposite to point <span class="texhtml mvar" style="font-style:italic;">V</span>. Angles <span class="texhtml">∠<i>DVE</i>, ∠<i>EVC</i></span> are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them.

</p><p>Therefore,

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle DVC=\angle DVE+\angle EVC.}">

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<annotation encoding="application/x-tex">{\displaystyle \angle DVC=\angle DVE+\angle EVC.}</annotation>

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</p><p>then let

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi _{0}&=\angle DVC,\\\psi _{1}&=\angle DVE,\\\psi _{2}&=\angle EVC,\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi _{0}&=\angle DVC,\\\psi _{1}&=\angle DVE,\\\psi _{2}&=\angle EVC,\end{aligned}}}</annotation>

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</p><p>so that

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}=\psi _{1}+\psi _{2}.\qquad \qquad (1)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{0}=\psi _{1}+\psi _{2}.\qquad \qquad (1)}</annotation>

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</p><p>Draw lines <span class="texhtml mvar" style="font-style:italic;">OC</span> and <span class="texhtml mvar" style="font-style:italic;">OD</span>. Angle <span class="texhtml">∠<i>DOC</i></span> is a central angle, but so are angles <span class="texhtml">∠<i>DOE</i></span> and <span class="texhtml">∠<i>EOC</i></span>, and

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<annotation encoding="application/x-tex">{\displaystyle \angle DOC=\angle DOE+\angle EOC.}</annotation>

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</p><p>Let

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\theta _{0}&=\angle DOC,\\\theta _{1}&=\angle DOE,\\\theta _{2}&=\angle EOC,\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\theta _{0}&=\angle DOC,\\\theta _{1}&=\angle DOE,\\\theta _{2}&=\angle EOC,\end{aligned}}}</annotation>

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</p><p>so that

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{0}=\theta _{1}+\theta _{2}.\qquad \qquad (2)}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{0}=\theta _{1}+\theta _{2}.\qquad \qquad (2)}</annotation>

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</p><p>From Part One we know that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{1}=2\psi _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{1}=2\psi _{1}}</annotation>

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</math></span><img src="./151233baf39152c8ba2cfb933d99e1e498899a0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.973ex; height:2.509ex;" alt="{\displaystyle \theta _{1}=2\psi _{1}}" loading="lazy"></span> and that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{2}=2\psi _{2}}">

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

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</math></span><img src="./a23a6aa7b38516159a99cb85082d27119cb6b014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.973ex; height:2.509ex;" alt="{\displaystyle \theta _{2}=2\psi _{2}}" loading="lazy"></span>. Combining these results with equation (2) yields

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{0}=2\psi _{1}+2\psi _{2}=2(\psi _{1}+\psi _{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{0}=2\psi _{1}+2\psi _{2}=2(\psi _{1}+\psi _{2})}</annotation>

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</p><p>therefore, by equation (1),

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{0}=2\psi _{0}.}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{0}=2\psi _{0}.}</annotation>

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<div class="mw-heading mw-heading4"><h4 id="Inscribed_angles_with_the_center_of_the_circle_in_their_exterior">Inscribed angles with the center of the circle in their exterior</h4></div>

<p>The previous case can be extended to cover the case where the measure of the inscribed angle is the <i>difference</i> between two inscribed angles as discussed in the first part of this proof.

</p><p>Given a circle whose center is point <span class="texhtml mvar" style="font-style:italic;">O</span>, choose three points <span class="texhtml mvar" style="font-style:italic;">V, C, D</span> on the circle. Draw lines <span class="texhtml mvar" style="font-style:italic;">VC</span> and <span class="texhtml mvar" style="font-style:italic;">VD</span>: angle <span class="texhtml">∠<i>DVC</i></span> is an inscribed angle. Now draw line <span class="texhtml mvar" style="font-style:italic;">OV</span> and extend it past point <span class="texhtml mvar" style="font-style:italic;">O</span> so that it intersects the circle at point <span class="texhtml mvar" style="font-style:italic;">E</span>. Angle <span class="texhtml">∠<i>DVC</i></span> intercepts arc <span class="texhtml mvar" style="font-style:italic;"><span style="line-height: 1.2em; padding-top: 0.2em; border: 1px solid transparent; border-top-color: var(--color-base,#202122); border-top-left-radius: 50% 25%; border-top-right-radius: 50% 25%;">DC</span></span> on the circle.

</p><p>Suppose this arc does not include point <span class="texhtml mvar" style="font-style:italic;">E</span> within it. Point <span class="texhtml mvar" style="font-style:italic;">E</span> is diametrically opposite to point <span class="texhtml mvar" style="font-style:italic;">V</span>. Angles <span class="texhtml">∠<i>EVD</i>, ∠<i>EVC</i></span> are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them.

</p><p>Therefore,

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle DVC=\angle EVC-\angle EVD.}">

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</p><p>then let

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi _{0}&=\angle DVC,\\\psi _{1}&=\angle EVD,\\\psi _{2}&=\angle EVC,\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi _{0}&=\angle DVC,\\\psi _{1}&=\angle EVD,\\\psi _{2}&=\angle EVC,\end{aligned}}}</annotation>

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</p><p>so that

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}=\psi _{2}-\psi _{1}.\qquad \qquad (3)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{0}=\psi _{2}-\psi _{1}.\qquad \qquad (3)}</annotation>

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</p><p>Draw lines <span class="texhtml mvar" style="font-style:italic;">OC</span> and <span class="texhtml mvar" style="font-style:italic;">OD</span>. Angle <span class="texhtml">∠<i>DOC</i></span> is a central angle, but so are angles <span class="texhtml">∠<i>EOD</i></span> and <span class="texhtml">∠<i>EOC</i></span>, and

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle DOC=\angle EOC-\angle EOD.}">

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<annotation encoding="application/x-tex">{\displaystyle \angle DOC=\angle EOC-\angle EOD.}</annotation>

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</p><p>Let

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\theta _{0}&=\angle DOC,\\\theta _{1}&=\angle EOD,\\\theta _{2}&=\angle EOC,\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\theta _{0}&=\angle DOC,\\\theta _{1}&=\angle EOD,\\\theta _{2}&=\angle EOC,\end{aligned}}}</annotation>

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</p><p>so that

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{0}=\theta _{2}-\theta _{1}.\qquad \qquad (4)}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{0}=\theta _{2}-\theta _{1}.\qquad \qquad (4)}</annotation>

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</p><p>From Part One we know that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{1}=2\psi _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{1}=2\psi _{1}}</annotation>

</semantics>

</math></span><img src="./151233baf39152c8ba2cfb933d99e1e498899a0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.973ex; height:2.509ex;" alt="{\displaystyle \theta _{1}=2\psi _{1}}" loading="lazy"></span> and that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{2}=2\psi _{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a23a6aa7b38516159a99cb85082d27119cb6b014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.973ex; height:2.509ex;" alt="{\displaystyle \theta _{2}=2\psi _{2}}" loading="lazy"></span>. Combining these results with equation (4) yields

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{0}=2\psi _{2}-2\psi _{1}}</annotation>

</semantics>

</math></span></span>

therefore, by equation (3),

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{0}=2\psi _{0}.}</annotation>

</semantics>

</math></span></span>

</p><p><br>

</p>

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

<p>By a similar argument, the angle between a <a href="Chord_(geometry)" title="Chord (geometry)">chord</a> and the <a href="Tangent" title="Tangent">tangent</a> line at one of its intersection points equals half of the central angle subtended by the chord. See also <a href="Tangent_lines_to_circles" title="Tangent lines to circles">Tangent lines to circles</a>.

</p>

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

<p>The inscribed angle <a href="Theorem" title="Theorem">theorem</a> is used in many proofs of elementary <a href="Euclidean_geometry_of_the_plane" class="mw-redirect" title="Euclidean geometry of the plane">Euclidean geometry of the plane</a>. A special case of the theorem is <a href="Thales's_theorem" title="Thales's theorem">Thales's theorem</a>, which states that the angle subtended by a <a href="Diameter" title="Diameter">diameter</a> is always 90°, i.e., a right angle. As a consequence of the theorem, opposite angles of <a href="Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilaterals</a> sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in a circle. As another example, the inscribed angle theorem is the basis for several theorems related to the <a href="Power_of_a_point" title="Power of a point">power of a point</a> with respect to a circle. Further, it allows one to prove that when two chords intersect in a circle, the products of the lengths of their pieces are equal.

</p>

<div class="mw-heading mw-heading2"><h2 id="Inscribed_angle_theorems_for_ellipses,_hyperbolas_and_parabolas">Inscribed angle theorems for ellipses, hyperbolas and parabolas</h2></div>

<p>Inscribed angle theorems exist for ellipses, hyperbolas and parabolas too. The essential differences are the measurements of an angle. (An angle is considered a pair of intersecting lines.)

</p>

<ul><li><a href="Ellipse#Inscribed_angles_and_three-point_form" title="Ellipse">Ellipse</a></li>

<li><a href="Hyperbola#Inscribed_angles_for_hyperbolas_y_=_a/(x_−_b)_+_c_and_the_3-point-form" title="Hyperbola">Hyperbola</a></li>

<li><a href="Parabola#Inscribed_angles_and_the_3-point_form" title="Parabola">Parabola</a></li></ul>

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

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFOgilvy1990" class="citation book cs1"><a href="C._Stanley_Ogilvy" title="C. Stanley Ogilvy">Ogilvy, C. S.</a> (1990). <i>Excursions in Geometry</i>. Dover. pp. <span class="nowrap">17–</span>23. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-26530-7</bdi>.</cite></li>

<li><cite id="CITEREFGellertKüstnerHellwichKästner1977" class="citation book cs1">Gellert W, Küstner H, Hellwich M, Kästner H (1977). <i>The VNR Concise Encyclopedia of Mathematics</i>. New York: Van Nostrand Reinhold. p. 172. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-442-22646-2</bdi>.</cite></li>

<li><cite id="CITEREFMoise1974" class="citation book cs1"><a href="Edwin_E._Moise" title="Edwin E. Moise">Moise, Edwin E.</a> (1974). <i>Elementary Geometry from an Advanced Standpoint</i> (2nd ed.). Reading: Addison-Wesley. pp. <span class="nowrap">192–</span>197. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-04793-4</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="citation mathworld" id="Reference-Mathworld-Inscribed_Angle"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/InscribedAngle.html">"Inscribed Angle"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>

<li><a rel="nofollow" class="external text" href="http://www.mathalino.com/reviewer/plane-geometry/relationship-between-central-angle-and-inscribed-angle">Relationship Between Central Angle and Inscribed Angle</a></li>

<li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/pythagoras/Munching/inscribed.shtml">Munching on Inscribed Angles</a> at <a href="Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20061030174939/http://www.mathopenref.com/arccentralangle.html">Arc Central Angle</a> With interactive animation</li>

<li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/arcperipheralangle.html">Arc Peripheral (inscribed) Angle</a> With interactive animation</li>

<li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/arccentralangletheorem.html">Arc Central Angle Theorem</a> With interactive animation</li>

<li><a rel="nofollow" class="external text" href="https://bookofproofs.github.io/branches/geometry/elements-euclid/book--3-circles/inscribed-angle-theorem.html">At bookofproofs.github.io</a></li></ul>

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</style><div id="Ancient_Greek_mathematics550" style="font-size:114%;margin:0 4em"><a href="Ancient_Greek_mathematics" title="Ancient Greek mathematics">Ancient Greek mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_Greek_mathematicians" title="List of Greek mathematicians">Mathematicians</a><br><a href="Timeline_of_ancient_Greek_mathematicians" title="Timeline of ancient Greek mathematicians">(timeline)</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

<li><a href="Anthemius_of_Tralles" title="Anthemius of Tralles">Anthemius</a></li>

<li><a href="Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li>

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

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

<li><a href="Aristaeus_the_Elder" title="Aristaeus the Elder">Aristaeus the Elder</a></li>

<li><a href="Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus</a></li>

<li><a href="Autolycus_of_Pitane" title="Autolycus of Pitane">Autolycus</a></li>

<li><a href="Bryson_of_Heraclea" title="Bryson of Heraclea">Bryson</a></li>

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

<li><a href="Carpus_of_Antioch" title="Carpus of Antioch">Carpus</a></li>

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

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

<li><a href="Conon_of_Samos" title="Conon of Samos">Conon</a></li>

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

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

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

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

<li><a href="Diocles_(mathematician)" title="Diocles (mathematician)">Diocles</a></li>

<li><a href="Dionysodorus" title="Dionysodorus">Dionysodorus of Caunus</a></li>

<li><a href="Dionysodorus_of_Amisene" title="Dionysodorus of Amisene">Dionysodorus of Amisene</a></li>

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

<li><a href="Domninus_of_Larissa" title="Domninus of Larissa">Domninus</a></li>

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

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

<li><a href="Eudemus_of_Rhodes" title="Eudemus of Rhodes">Eudemus</a></li>

<li><a href="Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a></li>

<li><a href="Eutocius_of_Ascalon" title="Eutocius of Ascalon">Eutocius</a></li>

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

<li><a href="Heliodorus_of_Larissa" title="Heliodorus of Larissa">Heliodorus</a></li>

<li><a href="Hero_of_Alexandria" title="Hero of Alexandria">Heron</a></li>

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

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

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

<li><a href="Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates</a></li>

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

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

<li><a href="Isidore_of_Miletus" title="Isidore of Miletus">Isidore of Miletus</a></li>

<li><a href="Marinus_of_Neapolis" title="Marinus of Neapolis">Marinus</a></li>

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

<li><a href="Menelaus_of_Alexandria" title="Menelaus of Alexandria">Menelaus</a></li>

<li><a href="Metrodorus_(grammarian)" title="Metrodorus (grammarian)">Metrodorus</a></li>

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

<li><a href="Nicomedes_(mathematician)" title="Nicomedes (mathematician)">Nicomedes</a></li>

<li><a href="Nicoteles_of_Cyrene" title="Nicoteles of Cyrene">Nicoteles</a></li>

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

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

<li><a href="Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus</a></li>

<li><a href="Perseus_(geometer)" title="Perseus (geometer)">Perseus</a></li>

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

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

<li><a href="Philonides_of_Laodicea" title="Philonides of Laodicea">Philonides</a></li>

<li><a href="Porphyry_of_Tyre" title="Porphyry of Tyre">Porphyry of Tyre</a></li>

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

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

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

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

<li><a href="Serenus_of_Antino%C3%B6polis" title="Serenus of Antinoöpolis">Serenus</a></li>

<li><a href="Sosigenes_of_Alexandria" class="mw-redirect" title="Sosigenes of Alexandria">Sosigenes</a></li>

<li><a href="Sporus_of_Nicaea" title="Sporus of Nicaea">Sporus</a></li>

<li><a href="Thales_of_Miletus" title="Thales of Miletus">Thales</a></li>

<li><a href="Theaetetus_(mathematician)" title="Theaetetus (mathematician)">Theaetetus</a></li>

<li><a href="Theodorus_of_Cyrene" title="Theodorus of Cyrene">Theodorus</a></li>

<li><a href="Theodosius_of_Bithynia" title="Theodosius of Bithynia">Theodosius</a></li>

<li><a href="Theon_of_Alexandria" title="Theon of Alexandria">Theon of Alexandria</a></li>

<li><a href="Theon_of_Smyrna" title="Theon of Smyrna">Theon of Smyrna</a></li>

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

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

<li><a href="Zeno_of_Sidon" title="Zeno of Sidon">Zeno of Sidon</a></li>

<li><a href="Zenodorus_(mathematician)" title="Zenodorus (mathematician)">Zenodorus</a></li></ul>

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

<ul><li><i><a href="Almagest" title="Almagest">Almagest</a></i></li>

<li><i><a href="Arithmetica" class="mw-redirect" title="Arithmetica">Arithmetica</a></i></li>

<li><a href="Apollonius_of_Perga#Conics" title="Apollonius of Perga"><i>Conics</i> <span style="font-size: 85%;">(Apollonius)</span></a></li>

<li><i><a href="Catoptrics" title="Catoptrics">Catoptrics</a></i></li>

<li><a href="Data_(Euclid)" class="mw-redirect" title="Data (Euclid)"><i>Data</i> <span style="font-size: 85%;">(Euclid)</span></a></li>

<li><a href="Euclid's_Elements" title="Euclid's Elements"><i>Elements</i> <span style="font-size: 85%;">(Euclid)</span></a></li>

<li><i><a href="Little_Astronomy" title="Little Astronomy">Little Astronomy</a></i></li>

<li><i><a href="Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a></i></li>

<li><i><a href="On_Conoids_and_Spheroids" title="On Conoids and Spheroids">On Conoids and Spheroids</a></i></li>

<li><a href="On_the_Sizes_and_Distances_(Aristarchus)" title="On the Sizes and Distances (Aristarchus)"><i>On the Sizes and Distances</i> <span style="font-size: 85%;">(Aristarchus)</span></a></li>

<li><a href="On_Sizes_and_Distances_(Hipparchus)" title="On Sizes and Distances (Hipparchus)"><i>On Sizes and Distances</i> <span style="font-size: 85%;">(Hipparchus)</span></a></li>

<li><a href="Autolycus_of_Pitane" title="Autolycus of Pitane"><i>On the Moving Sphere</i> <span style="font-size: 85%;">(Autolycus)</span></a></li>

<li><a href="Euclid's_Optics" title="Euclid's Optics"><i>Optics</i> <span style="font-size: 85%;">(Euclid)</span></a></li>

<li><i><a href="On_Spirals" title="On Spirals">On Spirals</a></i></li>

<li><i><a href="On_the_Sphere_and_Cylinder" title="On the Sphere and Cylinder">On the Sphere and Cylinder</a></i></li>

<li><i><a href="Ostomachion" title="Ostomachion">Ostomachion</a></i></li>

<li><a href="Euclid's_Phaenomena" title="Euclid's Phaenomena"><i>Phaenomena</i> <span style="font-size: 85%;">(Euclid)</span></a></li>

<li><i><a href="Planisphaerium" title="Planisphaerium">Planisphaerium</a></i></li>

<li><a href="Theodosius'_Spherics" title="Theodosius' Spherics"><i>Spherics</i> <span style="font-size: 85%;">(Theodosius)</span></a></li>

<li><a href="Menelaus_of_Alexandria" title="Menelaus of Alexandria"><i>Spherics</i> <span style="font-size: 85%;">(Menelaus)</span></a></li>

<li><i><a href="The_Quadrature_of_the_Parabola" class="mw-redirect" title="The Quadrature of the Parabola">The Quadrature of the Parabola</a></i></li>

<li><i><a href="The_Sand_Reckoner" title="The Sand Reckoner">The Sand Reckoner</a></i></li></ul>

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

<ul><li><a href="Chord_(geometry)" title="Chord (geometry)">Chord</a></li>

<li><a href="Circles_of_Apollonius" title="Circles of Apollonius">Circles of Apollonius</a>

<ul><li><a href="Apollonian_circles" title="Apollonian circles">Apollonian circles</a></li>

<li><a href="Apollonian_gasket" title="Apollonian gasket">Apollonian gasket</a></li>

<li><a href="Problem_of_Apollonius" title="Problem of Apollonius">Problem of Apollonius</a></li></ul></li>

<li><a href="Commensurability_(mathematics)" title="Commensurability (mathematics)">Commensurability</a></li>

<li><a href="Diophantine_equation" title="Diophantine equation">Diophantine equation</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li>

<li><a href="Golden_ratio" title="Golden ratio">Golden ratio</a></li>

<li><a href="Lune_of_Hippocrates" title="Lune of Hippocrates">Lune of Hippocrates</a></li>

<li><a href="Method_of_exhaustion" title="Method of exhaustion">Method of exhaustion</a></li>

<li><a href="Parallel_postulate" title="Parallel postulate">Parallel postulate</a></li>

<li><a href="Platonic_solid" title="Platonic solid">Platonic solid</a></li>

<li><a href="Regular_polygon" title="Regular polygon">Regular polygon</a></li>

<li><a href="Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass construction</a>

<ul><li><a href="Angle_trisection" title="Angle trisection">Angle trisection</a></li>

<li><a href="Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li>

<li><a href="Squaring_the_circle" title="Squaring the circle">Squaring the circle</a></li>

<li><a href="Quadratrix_of_Hippias" title="Quadratrix of Hippias">Quadratrix of Hippias</a></li>

<li><a href="Neusis_construction" title="Neusis construction">Neusis construction</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="In_Elements37" scope="row" class="navbox-group" style="width:1%">In <a href="Euclid's_elements" class="mw-redirect" title="Euclid's elements"><i>Elements</i></a></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="Angle_bisector_theorem" title="Angle bisector theorem">Angle bisector theorem</a></li>

<li><a href="Exterior_angle_theorem" title="Exterior angle theorem">Exterior angle theorem</a></li>

<li><a href="Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a></li>

<li><a href="Euclid's_theorem" title="Euclid's theorem">Euclid's theorem</a></li>

<li><a href="Geometric_mean_theorem" title="Geometric mean theorem">Geometric mean theorem</a></li>

<li><a href="Hinge_theorem" title="Hinge theorem">Hinge theorem</a></li>

<li><a href="Inscribed_angle_theorem" class="mw-redirect" title="Inscribed angle theorem">Inscribed angle theorem</a></li>

<li><a href="Intercept_theorem" title="Intercept theorem">Intercept theorem</a></li>

<li><a href="Intersecting_chords_theorem" title="Intersecting chords theorem">Intersecting chords theorem</a></li>

<li><a href="Intersecting_secants_theorem" title="Intersecting secants theorem">Intersecting secants theorem</a></li>

<li><a href="Law_of_cosines" title="Law of cosines">Law of cosines</a></li>

<li><a href="Pons_asinorum" title="Pons asinorum">Pons asinorum</a></li>

<li><a href="Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li>

<li><a href="Tangent-secant_theorem" class="mw-redirect" title="Tangent-secant theorem">Tangent-secant theorem</a></li>

<li><a href="Thales's_theorem" title="Thales's theorem">Thales's theorem</a></li>

<li><a href="Theorem_of_the_gnomon" title="Theorem of the gnomon">Theorem of the gnomon</a></li></ul>

</div></td></tr></tbody></table><div>

<ul><li><a href="Apollonius's_theorem" title="Apollonius's theorem">Apollonius's theorem</a></li>

<li><a href="Aristarchus's_inequality" title="Aristarchus's inequality">Aristarchus's inequality</a></li>

<li><a href="Heron's_formula" title="Heron's formula">Heron's formula</a></li>

<li><a href="Law_of_sines" title="Law of sines">Law of sines</a></li>

<li><a href="Menelaus's_theorem" title="Menelaus's theorem">Menelaus's theorem</a></li>

<li><a href="Pappus's_area_theorem" title="Pappus's area theorem">Pappus's area theorem</a></li>

<li><a href="Diophantus_II.VIII" title="Diophantus II.VIII">Problem II.8 of <i>Arithmetica</i></a></li>

<li><a href="Ptolemy's_inequality" title="Ptolemy's inequality">Ptolemy's inequality</a></li>

<li><a href="Ptolemy's_table_of_chords" title="Ptolemy's table of chords">Ptolemy's table of chords</a></li>

<li><a href="Ptolemy's_theorem" title="Ptolemy's theorem">Ptolemy's theorem</a></li>

<li><a href="Spiral_of_Theodorus" title="Spiral of Theodorus">Spiral of Theodorus</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Centers/Schools</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cyrene%2C_Libya" title="Cyrene, Libya">Cyrene</a></li>

<li><a href="Platonic_Academy" title="Platonic Academy">Platonic Academy</a></li>

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

<li><a href="School_of_Chios" title="School of Chios">School of Chios</a></li>

</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 hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ancient_Greek_astronomy" title="Ancient Greek astronomy">Ancient Greek astronomy</a></li>

<li><a href="Attic_numerals" title="Attic numerals">Attic numerals</a></li>

<li><a href="Greek_numerals" title="Greek numerals">Greek numerals</a></li></ul>

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

<ul><li><i><a href="A_History_of_Greek_Mathematics" title="A History of Greek Mathematics">A History of Greek Mathematics</a></i>

<ul><li>by <a href="Thomas_Heath_(classicist)" title="Thomas Heath (classicist)">Thomas Heath</a></li></ul></li>

<li><a href="Archimedes_Palimpsest" title="Archimedes Palimpsest">Archimedes Palimpsest</a></li>

<li><a href="History_of_algebra" title="History of algebra">algebra</a>

<ul><li><a href="Timeline_of_algebra" title="Timeline of algebra">timeline</a></li></ul></li>

<li><a href="History_of_arithmetic" class="mw-redirect" title="History of arithmetic">arithmetic</a>

<ul><li><a href="Timeline_of_numerals_and_arithmetic" title="Timeline of numerals and arithmetic">timeline</a></li></ul></li>

<li><a href="History_of_calculus" title="History of calculus">calculus</a>

<ul><li><a href="Timeline_of_calculus_and_mathematical_analysis" title="Timeline of calculus and mathematical analysis">timeline</a></li></ul></li>

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

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

<li><a href="History_of_logic" title="History of logic">logic</a>

<ul><li><a href="Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li>

<li><a href="History_of_mathematics" title="History of mathematics">mathematics</a>

<ul><li><a href="Timeline_of_mathematics" title="Timeline of mathematics">timeline</a></li></ul></li>

<li><a href="History_of_numbers" class="mw-redirect" title="History of numbers">numbers</a>

<ul><li><a href="Prehistoric_counting" class="mw-redirect" title="Prehistoric counting">prehistoric counting</a></li></ul></li>

<li><a href="History_of_ancient_numeral_systems" title="History of ancient numeral systems">numeral systems</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other cultures</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="Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Arabian/Islamic</a></li>

<li><a href="Babylonian_mathematics" title="Babylonian mathematics">Babylonian</a></li>

<li><a href="Chinese_mathematics" title="Chinese mathematics">Chinese</a></li>

<li><a href="Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Egyptian</a></li>

<li><a href="Mathematics_of_the_Incas" title="Mathematics of the Incas">Incan</a></li>

<li><a href="Indian_mathematics" title="Indian mathematics">Indian</a></li>

<li><a href="Japanese_mathematics" title="Japanese mathematics">Japanese</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><span></span></span> </span><a href="Portal%3AAncient_Greece" title="Portal:Ancient Greece">Ancient Greece portal</a></b> • <b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></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 2025-02-24" href="https://en.wikipedia.org/wiki/?title=Inscribed_angle&oldid=1277476991">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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