Неравенство Птолемея

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<link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Неравенство_Птолемея rootpage-Неравенство_Птолемея skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Неравенство Птолемея</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">Материал из Википедии — свободной энциклопедии</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Перейти к навигации</a> <a class="mw-jump-link" href="#searchInput">Перейти к поиску</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ru" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Ptolemy_Inequality.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Ptolemy_Inequality.svg/220px-Ptolemy_Inequality.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Ptolemy_Inequality.svg/330px-Ptolemy_Inequality.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Ptolemy_Inequality.svg/440px-Ptolemy_Inequality.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Если 4 точки не лежат на одной окружности, то все три <i>неравенства Птолемея</i> строгие.</figcaption></figure> <p><b>Неравенство Птолемея</b> — неравенство на 6 расстояний между четвёркой точек на плоскости. </p><p>Названо в честь позднеэллинистического математика <a href="/wiki/%D0%9A%D0%BB%D0%B0%D0%B2%D0%B4%D0%B8%D0%B9_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D0%B9" title="Клавдий Птолемей">Клавдия Птолемея</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="ru" dir="ltr"><h2 id="mw-toc-heading">Содержание</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Формулировка"><span class="tocnumber">1</span> <span class="toctext">Формулировка</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#О_других_доказательствах"><span class="tocnumber">2</span> <span class="toctext">О других доказательствах</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Следствия"><span class="tocnumber">3</span> <span class="toctext">Следствия</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Вариации_и_обобщения"><span class="tocnumber">4</span> <span class="toctext">Вариации и обобщения</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#См._также"><span class="tocnumber">5</span> <span class="toctext">См. также</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Примечания"><span class="tocnumber">6</span> <span class="toctext">Примечания</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Литература"><span class="tocnumber">7</span> <span class="toctext">Литература</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Формулировка"><span id=".D0.A4.D0.BE.D1.80.D0.BC.D1.83.D0.BB.D0.B8.D1.80.D0.BE.D0.B2.D0.BA.D0.B0"></span>Формулировка</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&veaction=edit&section=1" title="Редактировать раздел «Формулировка»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&action=edit&section=1" title="Редактировать код раздела «Формулировка»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Для любых точек <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C,D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684d01c09b12e5a28987c6127567daef29ee3b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.3ex; height:2.509ex;" alt="{\displaystyle A,B,C,D}"></span> плоскости выполнено неравенство </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AC\cdot BD\leq AB\cdot CD+BC\cdot AD,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>C</mi> <mo>⋅</mo> <mi>B</mi> <mi>D</mi> <mo>≤</mo> <mi>A</mi> <mi>B</mi> <mo>⋅</mo> <mi>C</mi> <mi>D</mi> <mo>+</mo> <mi>B</mi> <mi>C</mi> <mo>⋅</mo> <mi>A</mi> <mi>D</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC\cdot BD\leq AB\cdot CD+BC\cdot AD,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4d35d4c7af9f44a9dfa01a63c272dd08d7251f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.216ex; height:2.509ex;" alt="{\displaystyle AC\cdot BD\leq AB\cdot CD+BC\cdot AD,}"></span></dd></dl> <p>причём равенство достигается <a href="/wiki/%D0%A2%D0%BE%D0%B3%D0%B4%D0%B0_%D0%B8_%D1%82%D0%BE%D0%BB%D1%8C%D0%BA%D0%BE_%D1%82%D0%BE%D0%B3%D0%B4%D0%B0,_%D0%BA%D0%BE%D0%B3%D0%B4%D0%B0" class="mw-redirect" title="Тогда и только тогда, когда">тогда и только тогда, когда</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}"></span> — выпуклый <a href="/wiki/%D0%92%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%BD%D1%8B%D0%B9_%D1%87%D0%B5%D1%82%D1%8B%D1%80%D1%91%D1%85%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA" title="Вписанный четырёхугольник">вписанный четырёхугольник</a>, или точки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C,D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684d01c09b12e5a28987c6127567daef29ee3b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.3ex; height:2.509ex;" alt="{\displaystyle A,B,C,D}"></span> лежат на одной прямой. </p><p>Случай равенства также называется <b>тождеством Птолемея</b>. </p><p><br /> </p> <style data-mw-deduplicate="TemplateStyles:r137842454">.mw-parser-output .ts-Скрытый_блок{margin:0;overflow:hidden;border-collapse:collapse;box-sizing:border-box;font-size:95%}.mw-parser-output .ts-Скрытый_блок-title{text-align:center;font-weight:bold;line-height:1.6em;min-height:1.2em}.mw-parser-output .ts-Скрытый_блок 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.ts-Скрытый_блок-transparent .mw-collapsible-content{padding:.25em 0}.mw-parser-output .ts-Скрытый_блок-gray.ts-Скрытый_блок-rightHideLink .mw-collapsible-toggle{padding-right:1em}.mw-parser-output .ts-Скрытый_блок-transparent.ts-Скрытый_блок-rightHideLink .mw-collapsible-toggle{padding-right:0}.mw-parser-output .ts-Скрытый_блок-gray.ts-Скрытый_блок-leftHideLink .mw-collapsible-toggle{padding-left:1em}.mw-parser-output .ts-Скрытый_блок-transparent.ts-Скрытый_блок-leftHideLink .mw-collapsible-toggle{padding-left:0}.mw-parser-output .ts-Скрытый_блок-gray.ts-Скрытый_блок-rightHideLink .ts-Скрытый_блок-title-leftTitle{padding-left:1em}.mw-parser-output .ts-Скрытый_блок-gray.ts-Скрытый_блок-leftHideLink .ts-Скрытый_блок-title-leftTitle{padding-left:6.5em}.mw-parser-output .ts-Скрытый_блок-gray.ts-Скрытый_блок-leftHideLink .ts-Скрытый_блок-title-rightTitle{padding-right:1em}.mw-parser-output .ts-Скрытый_блок-transparent.ts-Скрытый_блок-rightHideLink .ts-Скрытый_блок-title-rightTitle,.mw-parser-output .ts-Скрытый_блок-transparent.ts-Скрытый_блок-rightHideLink .ts-Скрытый_блок-title-leftTitle{padding-left:0}.mw-parser-output .ts-Скрытый_блок-transparent.ts-Скрытый_блок-leftHideLink .ts-Скрытый_блок-title-rightTitle,.mw-parser-output .ts-Скрытый_блок-transparent.ts-Скрытый_блок-leftHideLink .ts-Скрытый_блок-title-leftTitle{padding-right:0}.mw-parser-output .ts-Скрытый_блок+.ts-Скрытый_блок,.mw-parser-output .ts-Скрытый_блок+link+.ts-Скрытый_блок{border-top-style:hidden}</style><div class="mw-collapsible mw-collapsed ts-Скрытый_блок ts-Скрытый_блок-gray ts-Скрытый_блок-rightHideLink" style=""><div class="ts-Скрытый_блок-title" style="text-align: left;">Доказательство<div class="mw-collapsible-toggle-placeholder"></div></div><div class="mw-collapsible-content" style="text-align: left;"> <p>Простейшее доказательство получается с использованием комплексных чисел. Пусть комплексные числа <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1},\,z_{2},\,z_{3},\,z_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1},\,z_{2},\,z_{3},\,z_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb3eae26db6065938b1084d472728c5685d65b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.805ex; height:2.009ex;" alt="{\displaystyle z_{1},\,z_{2},\,z_{3},\,z_{4}}"></span> соответствуют точкам <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C,D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684d01c09b12e5a28987c6127567daef29ee3b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.3ex; height:2.509ex;" alt="{\displaystyle A,B,C,D}"></span> плоскости. Тогда неравенство Птолемея равносильно неравенству </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z_{3}-z_{1}|\cdot |z_{4}-z_{2}|\leq |z_{3}-z_{2}|\cdot |z_{4}-z_{1}|+|z_{3}-z_{4}|\cdot |z_{1}-z_{2}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z_{3}-z_{1}|\cdot |z_{4}-z_{2}|\leq |z_{3}-z_{2}|\cdot |z_{4}-z_{1}|+|z_{3}-z_{4}|\cdot |z_{1}-z_{2}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8edab503723e57d47678acf21c773d5fcc859d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.405ex; height:2.843ex;" alt="{\displaystyle |z_{3}-z_{1}|\cdot |z_{4}-z_{2}|\leq |z_{3}-z_{2}|\cdot |z_{4}-z_{1}|+|z_{3}-z_{4}|\cdot |z_{1}-z_{2}|}"></span>, <br /></dd></dl> <p>которое следует из неравенства треугольника для комплексных чисел и тождества </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z_{3}-z_{1})\cdot (z_{4}-z_{2})=(z_{3}-z_{2})\cdot (z_{4}-z_{1})+(z_{3}-z_{4})\cdot (z_{1}-z_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (z_{3}-z_{1})\cdot (z_{4}-z_{2})=(z_{3}-z_{2})\cdot (z_{4}-z_{1})+(z_{3}-z_{4})\cdot (z_{1}-z_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3e75d82305a49d50d811fa8f65e65d40358e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.499ex; height:2.843ex;" alt="{\displaystyle (z_{3}-z_{1})\cdot (z_{4}-z_{2})=(z_{3}-z_{2})\cdot (z_{4}-z_{1})+(z_{3}-z_{4})\cdot (z_{1}-z_{2})}"></span>.</dd></dl> <p>В случае обращения неравенства в равенство слагаемые в правой части должны быть пропорциональны вектору суммы и сонаправлены ему, то есть оба числа </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {(z_{3}-z_{2})(z_{4}-z_{1})}{(z_{3}-z_{1})(z_{4}-z_{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {(z_{3}-z_{2})(z_{4}-z_{1})}{(z_{3}-z_{1})(z_{4}-z_{2})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a4b0c112e0221249d33a54039b455e478d03a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.677ex; height:7.176ex;" alt="{\displaystyle {\cfrac {(z_{3}-z_{2})(z_{4}-z_{1})}{(z_{3}-z_{1})(z_{4}-z_{2})}}}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {(z_{3}-z_{4})(z_{1}-z_{2})}{(z_{3}-z_{1})(z_{4}-z_{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {(z_{3}-z_{4})(z_{1}-z_{2})}{(z_{3}-z_{1})(z_{4}-z_{2})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c9de7de7a8daf305b88e91be6fb7534658c27b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.677ex; height:7.176ex;" alt="{\displaystyle {\cfrac {(z_{3}-z_{4})(z_{1}-z_{2})}{(z_{3}-z_{1})(z_{4}-z_{2})}}}"></span></dd></dl> <p>должны быть вещественны, положительны, с суммой равной 1 (то есть находятся между 0 и 1). </p><p>Вещественность означает, что <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {Im}}{\cfrac {z_{3}-z_{2}}{z_{3}-z_{1}}}:{\cfrac {z_{4}-z_{2}}{z_{4}-z_{1}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {Im}}{\cfrac {z_{3}-z_{2}}{z_{3}-z_{1}}}:{\cfrac {z_{4}-z_{2}}{z_{4}-z_{1}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/607255210413a56a344b7931b06459b58abbcba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.868ex; height:7.176ex;" alt="{\displaystyle {\rm {Im}}{\cfrac {z_{3}-z_{2}}{z_{3}-z_{1}}}:{\cfrac {z_{4}-z_{2}}{z_{4}-z_{1}}}=0}"></span>, а это - стандартное уравнение окружности. </p><p>То, что числа между 0 и 1 означает лишь, что точки A и C на этой окружности - не соседние (на обеих дугах между ними должна присутствовать либо точка B, либо точка D). </p> </div></div> <div class="mw-heading mw-heading2"><h2 id="О_других_доказательствах"><span id=".D0.9E_.D0.B4.D1.80.D1.83.D0.B3.D0.B8.D1.85_.D0.B4.D0.BE.D0.BA.D0.B0.D0.B7.D0.B0.D1.82.D0.B5.D0.BB.D1.8C.D1.81.D1.82.D0.B2.D0.B0.D1.85"></span>О других доказательствах</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&veaction=edit&section=2" title="Редактировать раздел «О других доказательствах»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&action=edit&section=2" title="Редактировать код раздела «О других доказательствах»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Один из вариантов доказательства неравенства основан на применении <a href="/wiki/%D0%98%D0%BD%D0%B2%D0%B5%D1%80%D1%81%D0%B8%D1%8F_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Инверсия (геометрия)">инверсии</a> относительно окружности с центром в точке <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>; этим неравенство Птолемея сводится к <a href="/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA%D0%B0" title="Неравенство треугольника">неравенству треугольника</a> для образов точек <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>Существует способ доказательства через <a href="/wiki/%D0%9F%D1%80%D1%8F%D0%BC%D0%B0%D1%8F_%D0%A1%D0%B8%D0%BC%D1%81%D0%BE%D0%BD%D0%B0" title="Прямая Симсона">прямую Симсона</a>.</li> <li>Теорема Птолемея может доказываться следующим способом (близким к доказательству самого Птолемея, приведённому им в книге <a href="/wiki/%D0%90%D0%BB%D1%8C%D0%BC%D0%B0%D0%B3%D0%B5%D1%81%D1%82" title="Альмагест">Альмагест</a>) — ввести точку <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> такую, что <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle ABE=\angle DBC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>E</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>D</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ABE=\angle DBC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4849392ead8c11476b3660137e3f3292a0a37dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.192ex; height:2.176ex;" alt="{\displaystyle \angle ABE=\angle DBC}"></span>, а потом через <a href="/wiki/%D0%9F%D0%BE%D0%B4%D0%BE%D0%B1%D0%B8%D0%B5_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA%D0%BE%D0%B2" class="mw-redirect" title="Подобие треугольников">подобие треугольников</a>.</li> <li>Теорема также является следствием из <a href="/wiki/%D0%A1%D0%BE%D0%BE%D1%82%D0%BD%D0%BE%D1%88%D0%B5%D0%BD%D0%B8%D0%B5_%D0%91%D1%80%D0%B5%D1%82%D1%88%D0%BD%D0%B0%D0%B9%D0%B4%D0%B5%D1%80%D0%B0" title="Соотношение Бретшнайдера">соотношения Бретшнайдера</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Следствия"><span id=".D0.A1.D0.BB.D0.B5.D0.B4.D1.81.D1.82.D0.B2.D0.B8.D1.8F"></span>Следствия</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&veaction=edit&section=3" title="Редактировать раздел «Следствия»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&action=edit&section=3" title="Редактировать код раздела «Следствия»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D0%BE%D0%BC%D0%BF%D0%B5%D1%8E" title="Теорема Помпею">Теорема Помпею</a></b>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Рассмотрим точку <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> и <a href="/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA" title="Правильный треугольник">правильный треугольник</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span>. Тогда из отрезков <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81e96775e04149546dbef15bc5152b07e158d97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.723ex; height:2.176ex;" alt="{\displaystyle XA}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa0a3fca929c224fde829e24ea00c0b099ddd8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.744ex; height:2.176ex;" alt="{\displaystyle XB}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab722fbd2027d28d36dc9931112013d87a5f812f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.746ex; height:2.176ex;" alt="{\displaystyle XC}"></span> можно составить треугольник, причём этот <a href="/wiki/%D0%A2%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA" title="Треугольник">треугольник</a> вырожденный тогда и только тогда, когда точка <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> лежит на описанной окружности треугольника <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span>.</li></ul> <ul><li>Если AC — <a href="/wiki/%D0%94%D0%B8%D0%B0%D0%BC%D0%B5%D1%82%D1%80" title="Диаметр">диаметр</a> окружности, то теорема превращается в правило <a href="/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81_(%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F)" class="mw-redirect" title="Синус (функция)">синуса</a> суммы. Именно это следствие использовал Птолемей для составления таблицы синусов.</li></ul> <ul><li><a href="/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%9A%D0%B0%D1%80%D0%BD%D0%BE" title="Формула Карно">Формула Карно</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Вариации_и_обобщения"><span id=".D0.92.D0.B0.D1.80.D0.B8.D0.B0.D1.86.D0.B8.D0.B8_.D0.B8_.D0.BE.D0.B1.D0.BE.D0.B1.D1.89.D0.B5.D0.BD.D0.B8.D1.8F"></span>Вариации и обобщения</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&veaction=edit&section=4" title="Редактировать раздел «Вариации и обобщения»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&action=edit&section=4" title="Редактировать код раздела «Вариации и обобщения»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D0%A1%D0%BE%D0%BE%D1%82%D0%BD%D0%BE%D1%88%D0%B5%D0%BD%D0%B8%D0%B5_%D0%91%D1%80%D0%B5%D1%82%D1%88%D0%BD%D0%B0%D0%B9%D0%B4%D0%B5%D1%80%D0%B0" title="Соотношение Бретшнайдера">Соотношение Бретшнайдера</a></li> <li>Неравенства Птолемея можно распространить и на шесть точек: если <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},A_{2},\dots A_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},A_{2},\dots A_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/876d601ca885f40e634c23231063de0f9b101f17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.57ex; height:2.509ex;" alt="{\displaystyle A_{1},A_{2},\dots A_{6}}"></span> произвольные точки плоскости (это обобщение называют <i>теоремой Птолемея для шестиугольника</i>, а в зарубежной литературе <i>теоремой Фурмана</i> (Fuhrmann’s theorem)<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>), то</li></ul> <dl><dd><dl><dd><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:%D0%9E%D0%B1%D0%BE%D0%B1%D1%89%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/%D0%9E%D0%B1%D0%BE%D0%B1%D1%89%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F.png/220px-%D0%9E%D0%B1%D0%BE%D0%B1%D1%89%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F.png" decoding="async" width="220" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/%D0%9E%D0%B1%D0%BE%D0%B1%D1%89%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F.png/330px-%D0%9E%D0%B1%D0%BE%D0%B1%D1%89%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F.png 1.5x, //upload.wikimedia.org/wikipedia/commons/9/97/%D0%9E%D0%B1%D0%BE%D0%B1%D1%89%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F.png 2x" data-file-width="407" data-file-height="430" /></a><figcaption>Обобщенная теорема Птолемея или <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9A%D0%B5%D0%B9%D1%81%D0%B8" title="Теорема Кейси">теорема Кейси</a></figcaption></figure><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}A_{4}\cdot A_{2}A_{5}\cdot A_{3}A_{6}\leq A_{1}A_{2}\cdot A_{3}A_{6}\cdot A_{4}A_{5}+A_{1}A_{2}\cdot A_{3}A_{4}\cdot A_{5}A_{6}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}A_{4}\cdot A_{2}A_{5}\cdot A_{3}A_{6}\leq A_{1}A_{2}\cdot A_{3}A_{6}\cdot A_{4}A_{5}+A_{1}A_{2}\cdot A_{3}A_{4}\cdot A_{5}A_{6}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837856d320b81516200711bac5bc4642430a108a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:68.174ex; height:2.509ex;" alt="{\displaystyle A_{1}A_{4}\cdot A_{2}A_{5}\cdot A_{3}A_{6}\leq A_{1}A_{2}\cdot A_{3}A_{6}\cdot A_{4}A_{5}+A_{1}A_{2}\cdot A_{3}A_{4}\cdot A_{5}A_{6}+}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +A_{2}A_{3}\cdot A_{1}A_{4}\cdot A_{5}A_{6}+A_{2}A_{3}\cdot A_{4}A_{5}\cdot A_{1}A_{6}+A_{3}A_{4}\cdot A_{2}A_{5}\cdot A_{1}A_{6},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +A_{2}A_{3}\cdot A_{1}A_{4}\cdot A_{5}A_{6}+A_{2}A_{3}\cdot A_{4}A_{5}\cdot A_{1}A_{6}+A_{3}A_{4}\cdot A_{2}A_{5}\cdot A_{1}A_{6},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f9d7467097be5b35be0b548c9aa5eb04408b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:68.563ex; height:2.509ex;" alt="{\displaystyle +A_{2}A_{3}\cdot A_{1}A_{4}\cdot A_{5}A_{6}+A_{2}A_{3}\cdot A_{4}A_{5}\cdot A_{1}A_{6}+A_{3}A_{4}\cdot A_{2}A_{5}\cdot A_{1}A_{6},}"></span></dd></dl></dd> <dd>причем равенство достигается тогда и только тогда, когда <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}\dots A_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}\dots A_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6a16e6bc712ddd245919df726af6663a7239bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.092ex; height:2.509ex;" alt="{\displaystyle A_{1}\dots A_{6}}"></span> — вписанный шестиугольник.</dd></dl> <ul><li><b><a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9A%D0%B5%D0%B9%D1%81%D0%B8" title="Теорема Кейси">Теорема Кейси</a></b> (<b>обобщённая теорема Птолемея</b>): Рассмотрим окружности <span class="mwe-math-element"><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 ,\beta ,\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791" 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 }"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span>, касающиеся данной окружности в вершинах <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> выпуклого четырёхугольника <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}"></span>. Пусть <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\alpha \beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f46dc6dd7c95b79255f2fa3f677cd3152c5feb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.066ex; height:2.676ex;" alt="{\displaystyle t_{\alpha \beta }}"></span> — длина общей касательной к окружностям <span class="mwe-math-element"><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</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" 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 }"></span> и <span class="mwe-math-element"><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</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" 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 }"></span> (внешней, если оба касания внутренние или внешние одновременно, и внутренней, если одно касание внутреннее, а другое внешнее); <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\beta \gamma },t_{\gamma \delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\beta \gamma },t_{\gamma \delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48459ca50cfbcc5e46ff554b7de871f27190e402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.646ex; height:2.676ex;" alt="{\displaystyle t_{\beta \gamma },t_{\gamma \delta }}"></span> <span class="nowrap">и т. д.</span> определяются аналогично. Тогда</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\alpha \beta }t_{\gamma \delta }+t_{\beta \gamma }t_{\delta \alpha }=t_{\alpha \gamma }t_{\beta \delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>γ</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\alpha \beta }t_{\gamma \delta }+t_{\beta \gamma }t_{\delta \alpha }=t_{\alpha \gamma }t_{\beta \delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af68a992c11309905a04da2a706c15af50e48aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.254ex; height:2.676ex;" alt="{\displaystyle t_{\alpha \beta }t_{\gamma \delta }+t_{\beta \gamma }t_{\delta \alpha }=t_{\alpha \gamma }t_{\beta \delta }}"></span>.</dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Circle_graph_C4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Circle_graph_C4.svg/170px-Circle_graph_C4.svg.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Circle_graph_C4.svg/255px-Circle_graph_C4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Circle_graph_C4.svg/340px-Circle_graph_C4.svg.png 2x" data-file-width="126" data-file-height="126" /></a><figcaption><a href="/wiki/%D0%A6%D0%B8%D0%BA%D0%BB%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B3%D1%80%D0%B0%D1%84" class="mw-redirect" title="Циклический граф">Циклический граф</a>, в котором все расстояния удовлетворяют <i>неравенству Птолемея</i>, называют <b>графом Птолемея</b></figcaption></figure> <ul><li><b><a href="/wiki/%D0%93%D1%80%D0%B0%D1%84_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F" class="mw-redirect" title="Граф Птолемея">Граф Птолемея</a></b> (см. рис.)<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>,</li></ul> <div class="mw-heading mw-heading2"><h2 id="См._также"><span id=".D0.A1.D0.BC._.D1.82.D0.B0.D0.BA.D0.B6.D0.B5"></span>См. также</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&veaction=edit&section=5" title="Редактировать раздел «См. также»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&action=edit&section=5" title="Редактировать код раздела «См. также»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D0%BE%D0%BC%D0%BF%D0%B5%D1%8E" title="Теорема Помпею">Теорема Помпею</a></li> <li><a href="/wiki/%D0%A2%D0%BE%D1%87%D0%BA%D0%B0_%D0%9C%D0%B8%D0%BA%D0%B5%D0%BB%D1%8F" title="Точка Микеля">Теорема Микеля о шести окружностях</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Примечания"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D0.B5.D1.87.D0.B0.D0.BD.D0.B8.D1.8F"></span>Примечания</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&veaction=edit&section=6" title="Редактировать раздел «Примечания»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&action=edit&section=6" title="Редактировать код раздела «Примечания»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.mccme.ru/ask/qa/ptolemey.html">Доказательство теоремы Птолемея с помощью инверсии</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090526081133/http://www.mccme.ru/ask/qa/ptolemey.html">Архивная копия</a> от 26 мая 2009 на <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Дистанционный консультационный пункт по математике <a href="/wiki/%D0%9C%D0%A6%D0%9D%D0%9C%D0%9E" class="mw-redirect" title="МЦНМО">МЦНМО</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.mccme.ru/ask/qa/pomp.html">О теореме Д. Помпейю</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041217075812/http://www.mccme.ru/ask/qa/pomp.html">Архивная копия</a> от 17 декабря 2004 на <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Дистанционный консультационный пункт по математике <a href="/wiki/%D0%9C%D0%A6%D0%9D%D0%9C%D0%9E" class="mw-redirect" title="МЦНМО">МЦНМО</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation"><span lang="und"><a rel="nofollow" class="external text" href="http://www.mccme.ru/ask/qa/ptolemey.html">Теорема Птолемея</a></span><span class="hidden-ref" style="display:none;">  <small class="ref-info" style="cursor:help;" title="на неопределённом языке">(неопр.)</small></span>. Дата обращения: 17 мая 2011. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090526081133/http://www.mccme.ru/ask/qa/ptolemey.html">Архивировано</a> 26 мая 2009 года.</span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r141305934">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a::after,.mw-parser-output .id-lock-limited a::after,.mw-parser-output .id-lock-registration a::after,.mw-parser-output .id-lock-subscription a::after,.mw-parser-output .cs1-ws-icon a::after{content:"";width:1.1em;height:1.1em;display:inline-block;vertical-align:middle;background-position:center;background-repeat:no-repeat;background-size:contain}.mw-parser-output .id-lock-free.id-lock-free a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")}.mw-parser-output .id-lock-limited.id-lock-limited a::after,.mw-parser-output .id-lock-registration.id-lock-registration a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")}.mw-parser-output .id-lock-subscription.id-lock-subscription a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")}.mw-parser-output .cs1-ws-icon a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}html.skin-theme-clientpref-night .mw-parser-output .id-lock-free a::after,html.skin-theme-clientpref-night .mw-parser-output .id-lock-limited a::after,html.skin-theme-clientpref-night .mw-parser-output .id-lock-registration a::after,html.skin-theme-clientpref-night .mw-parser-output .id-lock-subscription a::after{filter:invert(1)hue-rotate(180deg)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}html.skin-theme-clientpref-os .mw-parser-output .id-lock-free a::after,html.skin-theme-clientpref-os .mw-parser-output .id-lock-limited a::after,html.skin-theme-clientpref-os .mw-parser-output .id-lock-registration a::after,html.skin-theme-clientpref-os .mw-parser-output .id-lock-subscription a::after{filter:invert(1)hue-rotate(180deg)}}</style><cite id="CITEREFHoworka1981" class="citation cs2">Howorka, Edward (1981), "A characterization of Ptolemaic graphs (Характеризация графов Птолемея)", <i>Journal of Graph Theory</i>, <b>5</b> (3): 323—331, <a href="/wiki/%D0%A6%D0%B8%D1%84%D1%80%D0%BE%D0%B2%D0%BE%D0%B9_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%82%D0%BE%D1%80_%D0%BE%D0%B1%D1%8A%D0%B5%D0%BA%D1%82%D0%B0" title="Цифровой идентификатор объекта">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fjgt.3190050314">10.1002/jgt.3190050314</a>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0625074">0625074</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Graph+Theory&rft.atitle=A+characterization+of+Ptolemaic+graphs+%28%D0%A5%D0%B0%D1%80%D0%B0%D0%BA%D1%82%D0%B5%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%8F+%D0%B3%D1%80%D0%B0%D1%84%D0%BE%D0%B2+%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F%29&rft.volume=5&rft.issue=3&rft.pages=323%E2%80%94331&rft.date=1981&rft_id=info%3Adoi%2F10.1002%2Fjgt.3190050314&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D625074%23id-name%3DMR&rft.aulast=Howorka&rft.aufirst=Edward&rfr_id=info%3Asid%2Fru.wikipedia.org%3A%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE+%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F" class="Z3988"></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Литература"><span id=".D0.9B.D0.B8.D1.82.D0.B5.D1.80.D0.B0.D1.82.D1.83.D1.80.D0.B0"></span>Литература</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&veaction=edit&section=7" title="Редактировать раздел «Литература»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&action=edit&section=7" title="Редактировать код раздела «Литература»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343">Факультативный курс по математике. 7-9 / Сост. И. Л. Никольская. — <abbr title="Москва">М.</abbr>: <a href="/wiki/%D0%9F%D1%80%D0%BE%D1%81%D0%B2%D0%B5%D1%89%D0%B5%D0%BD%D0%B8%D0%B5_(%D0%B8%D0%B7%D0%B4%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D1%81%D1%82%D0%B2%D0%BE)" title="Просвещение (издательство)">Просвещение</a>, 1991. — С. 328-329. — 383 с. — <a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%98%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%B8_%D0%BA%D0%BD%D0%B8%D0%B3/5090012873" class="internal mw-magiclink-isbn">ISBN 5-09-001287-3</a>.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i><a href="/wiki/%D0%9F%D0%BE%D0%BD%D0%B0%D1%80%D0%B8%D0%BD,_%D0%AF%D0%BA%D0%BE%D0%B2_%D0%9F%D0%B5%D1%82%D1%80%D0%BE%D0%B2%D0%B8%D1%87" title="Понарин, Яков Петрович">Понарин Я. П.</a></i> Элементарная геометрия. В 2 т. — <abbr title="Москва">М.</abbr>: <a href="/wiki/%D0%9C%D0%A6%D0%9D%D0%9C%D0%9E" class="mw-redirect" title="МЦНМО">МЦНМО</a>, 2004. — С. 61-63. — <a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%98%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%B8_%D0%BA%D0%BD%D0%B8%D0%B3/5940571700" class="internal mw-magiclink-isbn">ISBN 5-94057-170-0</a>.</span></li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Источник — <a dir="ltr" href="https://ru.wikipedia.org/w/index.php?title=Неравенство_Птолемея&oldid=138135936">https://ru.wikipedia.org/w/index.php?title=Неравенство_Птолемея&oldid=138135936</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8" title="Служебная:Категории">Категории</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%9F%D0%BB%D0%B0%D0%BD%D0%B8%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Категория:Планиметрия">Планиметрия</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%B0" title="Категория:Неравенства">Неравенства</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D1%8B_%D0%BF%D0%BB%D0%B0%D0%BD%D0%B8%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B8" title="Категория:Теоремы планиметрии">Теоремы планиметрии</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Скрытые категории: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A1%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%8B,_%D0%B8%D1%81%D0%BF%D0%BE%D0%BB%D1%8C%D0%B7%D1%83%D1%8E%D1%89%D0%B8%D0%B5_%D1%80%D0%B0%D1%81%D1%88%D0%B8%D1%80%D0%B5%D0%BD%D0%B8%D0%B5_JsonConfig" title="Категория:Страницы, использующие расширение JsonConfig">Страницы, использующие расширение JsonConfig</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A1%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%8B,_%D0%B8%D1%81%D0%BF%D0%BE%D0%BB%D1%8C%D0%B7%D1%83%D1%8E%D1%89%D0%B8%D0%B5_%D0%B2%D0%BE%D0%BB%D1%88%D0%B5%D0%B1%D0%BD%D1%8B%D0%B5_%D1%81%D1%81%D1%8B%D0%BB%D0%BA%D0%B8_ISBN" title="Категория:Страницы, использующие волшебные ссылки ISBN">Страницы, использующие волшебные ссылки ISBN</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Навигация</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Персональные инструменты</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Страница участника для моего IP">Вы не представились системе</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9C%D0%BE%D1%91_%D0%BE%D0%B1%D1%81%D1%83%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Страница обсуждений для моего IP [n]" accesskey="n"><span>Обсуждение</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9C%D0%BE%D0%B9_%D0%B2%D0%BA%D0%BB%D0%B0%D0%B4" title="Список правок, сделанных с этого IP-адреса [y]" accesskey="y"><span>Вклад</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%BE%D0%B7%D0%B4%D0%B0%D1%82%D1%8C_%D1%83%D1%87%D1%91%D1%82%D0%BD%D1%83%D1%8E_%D0%B7%D0%B0%D0%BF%D0%B8%D1%81%D1%8C&returnto=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE+%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F" title="Мы предлагаем вам создать учётную запись и войти в систему, хотя это и не обязательно."><span>Создать учётную запись</span></a></li><li id="pt-login" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%92%D1%85%D0%BE%D0%B4&returnto=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE+%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F" title="Здесь можно зарегистрироваться в системе, но это необязательно. [o]" accesskey="o"><span>Войти</span></a></li> </ul> </div> </nav> <div id="left-navigation"> <nav id="p-namespaces" class="mw-portlet mw-portlet-namespaces vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-namespaces-label" > <h3 id="p-namespaces-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Пространства имён</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected mw-list-item"><a href="/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F" title="Просмотреть контентную страницу [c]" accesskey="c"><span>Статья</span></a></li><li id="ca-talk" class="mw-list-item"><a href="/wiki/%D0%9E%D0%B1%D1%81%D1%83%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5:%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F" rel="discussion" title="Обсуждение основной страницы [t]" accesskey="t"><span>Обсуждение</span></a></li> </ul> </div> </nav> <nav id="p-variants" class="mw-portlet mw-portlet-variants emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-variants-label" > <input type="checkbox" id="p-variants-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-variants" class="vector-menu-checkbox" aria-labelledby="p-variants-label" > <label id="p-variants-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">русский</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> </div> <div id="right-navigation"> <nav id="p-views" class="mw-portlet mw-portlet-views vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-views-label" > <h3 id="p-views-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Просмотры</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li 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title="Журнал изменений страницы [h]" accesskey="h"><span>История</span></a></li> </ul> </div> </nav> <nav id="p-cactions" class="mw-portlet mw-portlet-cactions emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-cactions-label" title="Больше возможностей" > <input type="checkbox" id="p-cactions-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-cactions" class="vector-menu-checkbox" aria-labelledby="p-cactions-label" > <label id="p-cactions-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Ещё</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <h3 >Поиск</h3> <form action="/w/index.php" id="searchform" class="vector-search-box-form"> <div id="simpleSearch" class="vector-search-box-inner" data-search-loc="header-navigation"> <input class="vector-search-box-input" type="search" name="search" placeholder="Искать в Википедии" aria-label="Искать в Википедии" autocapitalize="sentences" title="Искать в Википедии [f]" accesskey="f" id="searchInput" > <input type="hidden" name="title" value="Служебная:Поиск"> <input id="mw-searchButton" class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Найти страницы, содержащие указанный текст" value="Найти"> <input id="searchButton" class="searchButton" type="submit" name="go" title="Перейти к странице, имеющей в точности такое название" value="Перейти"> </div> </form> </div> </div> </div> <div id="mw-panel" class="vector-legacy-sidebar"> <div id="p-logo" role="banner"> <a class="mw-wiki-logo" href="/wiki/%D0%97%D0%B0%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Перейти на заглавную страницу"></a> </div> <nav id="p-navigation" class="mw-portlet 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Справки"><span>Справка</span></a></li> </ul> </div> </nav> <nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label" > <h3 id="p-tb-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Инструменты</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D1%81%D1%8B%D0%BB%D0%BA%D0%B8_%D1%81%D1%8E%D0%B4%D0%B0/%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F" title="Список всех страниц, ссылающихся на данную [j]" accesskey="j"><span>Ссылки сюда</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a 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id="p-coll-print_export-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Печать/экспорт</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:DownloadAsPdf&page=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&action=show-download-screen" title="Скачать эту страницу как файл PDF"><span>Скачать как PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F&printable=yes" title="Версия этой страницы для печати [p]" accesskey="p"><span>Версия для печати</span></a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">В других проектах</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Ptolemy%27s_theorem" hreflang="en"><span>Викисклад</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q459547" title="Ссылка на связанный элемент репозитория данных [g]" accesskey="g"><span>Элемент Викиданных</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">На других языках</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%A8%D8%B7%D9%84%D9%8A%D9%85%D9%88%D8%B3" title="مبرهنة بطليموس — арабский" lang="ar" hreflang="ar" data-title="مبرهنة بطليموس" data-language-autonym="العربية" data-language-local-name="арабский" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_Ptolemeu" title="Teorema de Ptolemeu — каталанский" lang="ca" hreflang="ca" data-title="Teorema de Ptolemeu" data-language-autonym="Català" data-language-local-name="каталанский" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%D9%85%DB%8C_%D8%A8%DB%95%D8%AA%DA%B5%DB%8C%D9%85%DB%86%D8%B3" title="تیۆرمی بەتڵیمۆس — центральнокурдский" lang="ckb" hreflang="ckb" data-title="تیۆرمی بەتڵیمۆس" data-language-autonym="کوردی" data-language-local-name="центральнокурдский" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_von_Ptolem%C3%A4us" title="Satz von Ptolemäus — немецкий" lang="de" hreflang="de" data-title="Satz von Ptolemäus" data-language-autonym="Deutsch" data-language-local-name="немецкий" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CF%84%CE%BF%CF%85_%CE%A0%CF%84%CE%BF%CE%BB%CE%B5%CE%BC%CE%B1%CE%AF%CE%BF%CF%85" title="Θεώρημα του Πτολεμαίου — греческий" lang="el" hreflang="el" data-title="Θεώρημα του Πτολεμαίου" data-language-autonym="Ελληνικά" data-language-local-name="греческий" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Ptolemy%27s_theorem" title="Ptolemy's theorem — английский" lang="en" hreflang="en" data-title="Ptolemy's theorem" data-language-autonym="English" data-language-local-name="английский" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_Ptolomeo" title="Teorema de Ptolomeo — испанский" lang="es" hreflang="es" data-title="Teorema de Ptolomeo" data-language-autonym="Español" data-language-local-name="испанский" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D8%A8%D8%B7%D9%84%D9%85%DB%8C%D9%88%D8%B3" title="قضیه بطلمیوس — персидский" lang="fa" hreflang="fa" data-title="قضیه بطلمیوس" data-language-autonym="فارسی" data-language-local-name="персидский" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ptolemaioksen_lause" title="Ptolemaioksen lause — финский" lang="fi" hreflang="fi" data-title="Ptolemaioksen lause" data-language-autonym="Suomi" data-language-local-name="финский" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Ptol%C3%A9m%C3%A9e" title="Théorème de Ptolémée — французский" lang="fr" hreflang="fr" data-title="Théorème de Ptolémée" data-language-autonym="Français" data-language-local-name="французский" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_de_Tolomeo" title="Teorema de Tolomeo — галисийский" lang="gl" hreflang="gl" data-title="Teorema de Tolomeo" data-language-autonym="Galego" data-language-local-name="галисийский" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%AA%D7%9C%D7%9E%D7%99" title="משפט תלמי — иврит" lang="he" hreflang="he" data-title="משפט תלמי" data-language-autonym="עברית" data-language-local-name="иврит" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%A4%E0%A5%8B%E0%A4%B2%E0%A5%87%E0%A4%AE%E0%A5%80_%E0%A4%95%E0%A4%BE_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="प्तोलेमी का प्रमेय — хинди" lang="hi" hreflang="hi" data-title="प्तोलेमी का प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="хинди" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ptolemejev_pou%C4%8Dak" title="Ptolemejev poučak — хорватский" lang="hr" hreflang="hr" data-title="Ptolemejev poučak" data-language-autonym="Hrvatski" data-language-local-name="хорватский" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ptolemaiosz-t%C3%A9tel" title="Ptolemaiosz-tétel — венгерский" lang="hu" hreflang="hu" data-title="Ptolemaiosz-tétel" data-language-autonym="Magyar" data-language-local-name="венгерский" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%BF%D5%B2%D5%B8%D5%B4%D5%A5%D5%B8%D5%BD%D5%AB_%D5%A9%D5%A5%D5%B8%D6%80%D5%A5%D5%B4" title="Պտղոմեոսի թեորեմ — армянский" lang="hy" hreflang="hy" data-title="Պտղոմեոսի թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="армянский" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_Tolomeo" title="Teorema di Tolomeo — итальянский" lang="it" hreflang="it" data-title="Teorema di Tolomeo" data-language-autonym="Italiano" data-language-local-name="итальянский" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86" title="トレミーの定理 — японский" lang="ja" hreflang="ja" data-title="トレミーの定理" data-language-autonym="日本語" data-language-local-name="японский" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%A2%E1%83%9D%E1%83%9A%E1%83%94%E1%83%9B%E1%83%94%E1%83%A1_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%94%E1%83%9B%E1%83%90" title="პტოლემეს თეორემა — грузинский" lang="ka" hreflang="ka" data-title="პტოლემეს თეორემა" data-language-autonym="ქართული" data-language-local-name="грузинский" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%91%E1%9F%92%E1%9E%9A%E1%9E%B9%E1%9E%9F%E1%9F%92%E1%9E%8F%E1%9E%B8%E1%9E%94%E1%9E%91%E1%9E%8F%E1%9E%BC%E1%9E%9B%E1%9F%81%E1%9E%98%E1%9E%B8" title="ទ្រឹស្តីបទតូលេមី — кхмерский" lang="km" hreflang="km" data-title="ទ្រឹស្តីបទតូលេមី" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="кхмерский" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%84%ED%86%A8%EB%A0%88%EB%A7%88%EC%9D%B4%EC%98%A4%EC%8A%A4_%EC%A0%95%EB%A6%AC" title="프톨레마이오스 정리 — корейский" lang="ko" hreflang="ko" data-title="프톨레마이오스 정리" data-language-autonym="한국어" data-language-local-name="корейский" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Ptolem%C4%97jaus_teorema" title="Ptolemėjaus teorema — литовский" lang="lt" hreflang="lt" data-title="Ptolemėjaus teorema" data-language-autonym="Lietuvių" data-language-local-name="литовский" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Ptolemaja_teor%C4%93ma" title="Ptolemaja teorēma — латышский" lang="lv" hreflang="lv" data-title="Ptolemaja teorēma" data-language-autonym="Latviešu" data-language-local-name="латышский" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D1%82%D0%BE%D0%BB%D0%BE%D0%BC%D0%B5%D0%B5%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Птоломеева теорема — македонский" lang="mk" hreflang="mk" data-title="Птоломеева теорема" data-language-autonym="Македонски" data-language-local-name="македонский" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Ptolemaeus" title="Stelling van Ptolemaeus — нидерландский" lang="nl" hreflang="nl" data-title="Stelling van Ptolemaeus" data-language-autonym="Nederlands" data-language-local-name="нидерландский" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ptolemaios-satsen" title="Ptolemaios-satsen — нюнорск" lang="nn" hreflang="nn" data-title="Ptolemaios-satsen" data-language-autonym="Norsk nynorsk" data-language-local-name="нюнорск" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Ptolemeusza" title="Twierdzenie Ptolemeusza — польский" lang="pl" hreflang="pl" data-title="Twierdzenie Ptolemeusza" data-language-autonym="Polski" data-language-local-name="польский" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Ptolomeu" title="Teorema de Ptolomeu — португальский" lang="pt" hreflang="pt" data-title="Teorema de Ptolomeu" data-language-autonym="Português" data-language-local-name="португальский" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorema_lui_Ptolemeu" title="Teorema lui Ptolemeu — румынский" lang="ro" hreflang="ro" data-title="Teorema lui Ptolemeu" data-language-autonym="Română" data-language-local-name="румынский" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ptolemajev_izrek" title="Ptolemajev izrek — словенский" lang="sl" hreflang="sl" data-title="Ptolemajev izrek" data-language-autonym="Slovenščina" data-language-local-name="словенский" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ptolemaios_sats" title="Ptolemaios sats — шведский" lang="sv" hreflang="sv" data-title="Ptolemaios sats" data-language-autonym="Svenska" data-language-local-name="шведский" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Uhakiki_wa_Ptolemaio" title="Uhakiki wa Ptolemaio — суахили" lang="sw" hreflang="sw" data-title="Uhakiki wa Ptolemaio" data-language-autonym="Kiswahili" data-language-local-name="суахили" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%8A%E0%AE%B2%E0%AF%86%E0%AE%AE%E0%AE%BF%E0%AE%AF%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="தொலெமியின் தேற்றம் — тамильский" lang="ta" hreflang="ta" data-title="தொலெமியின் தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="тамильский" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Batlamyus_teoremi" title="Batlamyus teoremi — турецкий" lang="tr" hreflang="tr" data-title="Batlamyus teoremi" data-language-autonym="Türkçe" data-language-local-name="турецкий" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D1%8F" title="Теорема Птолемея — украинский" lang="uk" hreflang="uk" data-title="Теорема Птолемея" data-language-autonym="Українська" data-language-local-name="украинский" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Ptoleme" title="Định lý Ptoleme — вьетнамский" lang="vi" hreflang="vi" data-title="Định lý Ptoleme" data-language-autonym="Tiếng Việt" data-language-local-name="вьетнамский" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%89%98%E5%8B%92%E5%AF%86%E5%AE%9A%E7%90%86" title="托勒密定理 — китайский" lang="zh" hreflang="zh" data-title="托勒密定理" data-language-autonym="中文" data-language-local-name="китайский" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%89%98%E5%8B%92%E5%AF%86%E5%AE%9A%E7%90%86" title="托勒密定理 — Literary Chinese" lang="lzh" hreflang="lzh" data-title="托勒密定理" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q459547#sitelinks-wikipedia" title="Править ссылки на другие языки" 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Неравенство Птолемея — Википедия