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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Halbwinkelsatz</span></h1>
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<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Halbwinkelsätze</b> sind Formeln der <a href="Trigonometrie" title="Trigonometrie">Trigonometrie</a>, die für spezielle, logarithmisch brauchbare Anwendungsfälle zur Ermittlung der Bestimmungsgrößen (Seiten a, b, c; Winkel <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 \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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"></span>, <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 \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.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"></span>, <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 \gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>γ<!-- γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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"></span>) von allgemeinen <a href="Dreieck" title="Dreieck">Dreiecken</a> entwickelt wurden. Entsprechende Sätze gelten für allgemeine Dreiecke auf einer <a href="Kugeloberfl%C3%A4che" class="mw-redirect" title="Kugeloberfläche">Kugeloberfläche</a> (<a href="Sph%C3%A4rische_Geometrie" title="Sphärische Geometrie">sphärische Geometrie</a>).
</p>
<div class="mw-heading mw-heading2"><h2 id="Halbwinkelsätze_in_der_Ebene"><span id="Halbwinkels.C3.A4tze_in_der_Ebene"></span>Halbwinkelsätze in der Ebene</h2></div>

<ul><li><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 \sin {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{bc}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>b</mi>
<mi>c</mi>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{bc}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3daf3f1a467f76953096397b6577fdf8d03e931.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.31ex; height:7.509ex;" alt="{\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{bc}}}}" loading="lazy"></span></li>
<li><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 \cos {\frac {\alpha }{2}}={\sqrt {\frac {s(s-a)}{bc}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>b</mi>
<mi>c</mi>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {s(s-a)}{bc}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e8ecf93f3cdfab4f4a1ea7d4d2536413ef578cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.141ex; height:7.509ex;" alt="{\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {s(s-a)}{bc}}}}" loading="lazy"></span></li>
<li><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 \tan {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{s(s-a)}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>tan</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{s(s-a)}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1311c22d696bfeb13158155d4a02ad406342af80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.814ex; height:7.509ex;" alt="{\displaystyle \tan {\frac {\alpha }{2}}={\sqrt {\frac {(s-b)(s-c)}{s(s-a)}}}}" loading="lazy"></span></li></ul>
<p>wobei <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 s={\frac {a+b+c}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s={\frac {a+b+c}{2}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/787a98dac5681f383514fc1bd5b4d8e561a3fd21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.94ex; height:5.343ex;" alt="{\displaystyle s={\frac {a+b+c}{2}}}" loading="lazy"></span>
</p><p>Die zur dritten Formel äquivalente Aussage
</p>
<ul><li><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 \cot {\frac {\alpha }{2}}={\frac {s-a}{\rho }}={\sqrt {\frac {s(s-a)}{(s-b)(s-c)}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cot</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
</mrow>
<mi>ρ<!-- ρ --></mi>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cot {\frac {\alpha }{2}}={\frac {s-a}{\rho }}={\sqrt {\frac {s(s-a)}{(s-b)(s-c)}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bb7ce0ea949545a5d1acdf89bb5d21feb3f1a73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.649ex; height:7.509ex;" alt="{\displaystyle \cot {\frac {\alpha }{2}}={\frac {s-a}{\rho }}={\sqrt {\frac {s(s-a)}{(s-b)(s-c)}}}}" loading="lazy"></span></li></ul>
<p>ist auch als <i>Kotangenssatz</i> bekannt. <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 \rho }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ρ<!-- ρ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> bezeichnet hier den
<a href="Inkreis" title="Inkreis">Inkreisradius</a>.
</p><p>Entsprechende Formeln gelten für die anderen Winkel.
</p>
<div class="mw-heading mw-heading2"><h2 id="Halbwinkelsätze_auf_der_Kugeloberfläche"><span id="Halbwinkels.C3.A4tze_auf_der_Kugeloberfl.C3.A4che"></span>Halbwinkelsätze auf der Kugeloberfläche</h2></div>
<ul><li><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 \sin {\frac {\alpha }{2}}={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin b\,\sin c}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>b</mi>
<mspace width="thinmathspace"></mspace>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>c</mi>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin b\,\sin c}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4af8ed25f437bca2deb067dc63200c569f789068.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.795ex; height:7.509ex;" alt="{\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin b\,\sin c}}}}" loading="lazy"></span></li>
<li><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 \cos {\frac {\alpha }{2}}={\sqrt {\frac {\sin s\,\sin(s-a)}{\sin b\,\sin c}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>s</mi>
<mspace width="thinmathspace"></mspace>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>b</mi>
<mspace width="thinmathspace"></mspace>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>c</mi>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {\sin s\,\sin(s-a)}{\sin b\,\sin c}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7284c9b00e61129e918f408c18a7d4da17f20a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.013ex; height:7.509ex;" alt="{\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {\sin s\,\sin(s-a)}{\sin b\,\sin c}}}}" loading="lazy"></span></li>
<li><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 \tan {\frac {\alpha }{2}}={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin s\,\sin(s-a)}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>tan</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>α<!-- α --></mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>s</mi>
<mspace width="thinmathspace"></mspace>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\alpha }{2}}={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin s\,\sin(s-a)}}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/341ef2efebec15338465db12507ce19c5affb22e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.299ex; height:7.509ex;" alt="{\displaystyle \tan {\frac {\alpha }{2}}={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin s\,\sin(s-a)}}}}" loading="lazy"></span></li></ul>
<p>wobei <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 s={\frac {a+b+c}{2}}}">
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<mi>s</mi>
<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle s={\frac {a+b+c}{2}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/787a98dac5681f383514fc1bd5b4d8e561a3fd21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.94ex; height:5.343ex;" alt="{\displaystyle s={\frac {a+b+c}{2}}}" loading="lazy"></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="Quellen">Quellen</h2></div>
<ul><li>Fachredaktion des <a href="Bibliographisches_Institut" title="Bibliographisches Institut">Bibliographischen Instituts</a> (Hrsg.): <cite style="font-style:italic">Duden Rechnen und Mathematik: Das Lexikon für Schule und Praxis</cite>. Bearbeitet von Prof. Dr. <a href="Harald_Scheid" title="Harald Scheid">Harald Scheid</a>. 4. Auflage. Bibliographisches Institut, Mannheim, Wien, Zürich 1985, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>622</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Halbwinkelsatz&amp;rft.btitle=Duden+Rechnen+und+Mathematik%3A+Das+Lexikon+f%C3%BCr+Schule+und+Praxis&amp;rft.date=1985&amp;rft.edition=4.&amp;rft.genre=book&amp;rft.pages=622&amp;rft.place=Mannheim%2C+Wien%2C+Z%C3%BCrich&amp;rft.pub=Bibliographisches+Institut" style="display:none">&nbsp;</span></li>
<li>F. Specht: <cite style="font-style:italic">Herleitung der trigonometrischen Formel für die Tangente des halben Winkels aus den Seiten des Dreiecks</cite>. In: <cite style="font-style:italic"><a href="Archiv_der_Mathematik_und_Physik" title="Archiv der Mathematik und Physik">Archiv der Mathematik und Physik</a>. 2. Reihe. Mit besonderer Rücksicht auf die Bedürfnisse der Lehrer an höheren Unterrichtsanstalten</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em">&nbsp;</span>XIII</span>, 1894, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>223–224</span> (<a rel="nofollow" class="external text" href="https://zbmath.org/25.0927.01">Eintrag <i>zbMATH Open</i></a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Halbwinkelsatz&amp;rft.atitle=Herleitung+der+trigonometrischen+Formel+f%C3%BCr+die+Tangente+des+halben+Winkels+aus+den+Seiten+des+Dreiecks&amp;rft.au=F.+Specht&amp;rft.btitle=Archiv+der+Mathematik+und+Physik.+2.+Reihe.+Mit+besonderer+R%C3%BCcksicht+auf+die+Bed%C3%BCrfnisse+der+Lehrer+an+h%C3%B6heren+Unterrichtsanstalten&amp;rft.date=1894&amp;rft.genre=book&amp;rft.pages=223-224&amp;rft.volume=XIII" style="display:none">&nbsp;</span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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