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href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%82%A2%E3%82%AB%E3%82%A6%E3%83%B3%E3%83%88%E4%BD%9C%E6%88%90&returnto=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86" title="アカウントを作成してログインすることをお勧めしますが、必須ではありません"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>アカウント作成</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%83%AD%E3%82%B0%E3%82%A4%E3%83%B3&returnto=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86" title="ログインすることを推奨します。ただし、必須ではありません。 [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>ログイン</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> ログアウトした編集者のページ <a href="/wiki/Wikipedia:%E3%82%A6%E3%82%A3%E3%82%AD%E3%83%9A%E3%83%87%E3%82%A3%E3%82%A2%E3%81%B8%E3%82%88%E3%81%86%E3%81%93%E3%81%9D" aria-label="編集の詳細"><span>もっと詳しく</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E8%87%AA%E5%88%86%E3%81%AE%E6%8A%95%E7%A8%BF%E8%A8%98%E9%8C%B2" title="このIPアドレスからなされた編集の一覧 [y]" accesskey="y"><span>投稿記録</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E3%83%88%E3%83%BC%E3%82%AF%E3%83%9A%E3%83%BC%E3%82%B8" title="このIPアドレスからなされた編集についての議論 [n]" accesskey="n"><span>トーク</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="サイト"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目次" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目次</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">非表示</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">ページ先頭</div> </a> </li> <li id="toc-証明" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#証明"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>証明</span> </div> </a> <button aria-controls="toc-証明-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>証明サブセクションを切り替えます</span> </button> <ul id="toc-証明-sublist" class="vector-toc-list"> <li id="toc-円に関する反転を用いた証明" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#円に関する反転を用いた証明"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>円に関する反転を用いた証明</span> </div> </a> <ul id="toc-円に関する反転を用いた証明-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-一般化" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#一般化"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>一般化</span> </div> </a> <ul id="toc-一般化-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-脚注" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#脚注"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>脚注</span> </div> </a> <ul id="toc-脚注-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-関連項目" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#関連項目"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>関連項目</span> </div> </a> <ul id="toc-関連項目-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部リンク" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部リンク"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>外部リンク</span> </div> </a> <ul id="toc-外部リンク-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目次" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="目次" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="目次の表示・非表示を切り替え" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">目次の表示・非表示を切り替え</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">トレミーの定理</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="特定の記事の別の言語版に移動します。利用可能な言語38件" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-38" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">38の言語版</span> </label> <div class="vector-dropdown-content"> <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="アラビア語: مبرهنة بطليموس" lang="ar" hreflang="ar" data-title="مبرهنة بطليموس" data-language-autonym="العربية" data-language-local-name="アラビア語" 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="カタロニア語: Teorema de Ptolemeu" lang="ca" hreflang="ca" data-title="Teorema de Ptolemeu" data-language-autonym="Català" data-language-local-name="カタロニア語" 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="中央クルド語: تیۆرمی بەتڵیمۆس" lang="ckb" hreflang="ckb" data-title="تیۆرمی بەتڵیمۆس" data-language-autonym="کوردی" data-language-local-name="中央クルド語" 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="ドイツ語: Satz von Ptolemäus" lang="de" hreflang="de" data-title="Satz von Ptolemäus" data-language-autonym="Deutsch" data-language-local-name="ドイツ語" 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="ギリシャ語: Θεώρημα του Πτολεμαίου" lang="el" hreflang="el" data-title="Θεώρημα του Πτολεμαίου" data-language-autonym="Ελληνικά" data-language-local-name="ギリシャ語" 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="英語: Ptolemy's theorem" lang="en" hreflang="en" data-title="Ptolemy's theorem" data-language-autonym="English" data-language-local-name="英語" 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="スペイン語: Teorema de Ptolomeo" lang="es" hreflang="es" data-title="Teorema de Ptolomeo" data-language-autonym="Español" data-language-local-name="スペイン語" 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="ペルシア語: قضیه بطلمیوس" lang="fa" hreflang="fa" data-title="قضیه بطلمیوس" data-language-autonym="فارسی" data-language-local-name="ペルシア語" 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="フィンランド語: Ptolemaioksen lause" lang="fi" hreflang="fi" data-title="Ptolemaioksen lause" data-language-autonym="Suomi" data-language-local-name="フィンランド語" 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="フランス語: Théorème de Ptolémée" lang="fr" hreflang="fr" data-title="Théorème de Ptolémée" data-language-autonym="Français" data-language-local-name="フランス語" 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="ガリシア語: Teorema de Tolomeo" lang="gl" hreflang="gl" data-title="Teorema de Tolomeo" data-language-autonym="Galego" data-language-local-name="ガリシア語" 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="ヘブライ語: משפט תלמי" lang="he" hreflang="he" data-title="משפט תלמי" data-language-autonym="עברית" data-language-local-name="ヘブライ語" 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="ヒンディー語: प्तोलेमी का प्रमेय" lang="hi" hreflang="hi" data-title="प्तोलेमी का प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="ヒンディー語" 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="クロアチア語: Ptolemejev poučak" lang="hr" hreflang="hr" data-title="Ptolemejev poučak" data-language-autonym="Hrvatski" data-language-local-name="クロアチア語" 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="ハンガリー語: Ptolemaiosz-tétel" lang="hu" hreflang="hu" data-title="Ptolemaiosz-tétel" data-language-autonym="Magyar" data-language-local-name="ハンガリー語" 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="アルメニア語: Պտղոմեոսի թեորեմ" lang="hy" hreflang="hy" data-title="Պտղոմեոսի թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="アルメニア語" 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="イタリア語: Teorema di Tolomeo" lang="it" hreflang="it" data-title="Teorema di Tolomeo" data-language-autonym="Italiano" data-language-local-name="イタリア語" class="interlanguage-link-target"><span>Italiano</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="ジョージア語: პტოლემეს თეორემა" lang="ka" hreflang="ka" data-title="პტოლემეს თეორემა" data-language-autonym="ქართული" data-language-local-name="ジョージア語" 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="クメール語: ទ្រឹស្តីបទតូលេមី" lang="km" hreflang="km" data-title="ទ្រឹស្តីបទតូលេមី" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="クメール語" 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="韓国語: 프톨레마이오스 정리" lang="ko" hreflang="ko" data-title="프톨레마이오스 정리" data-language-autonym="한국어" data-language-local-name="韓国語" 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="リトアニア語: Ptolemėjaus teorema" lang="lt" hreflang="lt" data-title="Ptolemėjaus teorema" data-language-autonym="Lietuvių" data-language-local-name="リトアニア語" 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="ラトビア語: Ptolemaja teorēma" lang="lv" hreflang="lv" data-title="Ptolemaja teorēma" data-language-autonym="Latviešu" data-language-local-name="ラトビア語" 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="マケドニア語: Птоломеева теорема" lang="mk" hreflang="mk" data-title="Птоломеева теорема" data-language-autonym="Македонски" data-language-local-name="マケドニア語" 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="オランダ語: Stelling van Ptolemaeus" lang="nl" hreflang="nl" data-title="Stelling van Ptolemaeus" data-language-autonym="Nederlands" data-language-local-name="オランダ語" 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="ノルウェー語(ニーノシュク): Ptolemaios-satsen" lang="nn" hreflang="nn" data-title="Ptolemaios-satsen" data-language-autonym="Norsk nynorsk" data-language-local-name="ノルウェー語(ニーノシュク)" 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="ポーランド語: Twierdzenie Ptolemeusza" lang="pl" hreflang="pl" data-title="Twierdzenie Ptolemeusza" data-language-autonym="Polski" data-language-local-name="ポーランド語" 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="ポルトガル語: Teorema de Ptolomeu" lang="pt" hreflang="pt" data-title="Teorema de Ptolomeu" data-language-autonym="Português" data-language-local-name="ポルトガル語" 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="ルーマニア語: Teorema lui Ptolemeu" lang="ro" hreflang="ro" data-title="Teorema lui Ptolemeu" data-language-autonym="Română" data-language-local-name="ルーマニア語" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/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="ロシア語: Неравенство Птолемея" lang="ru" hreflang="ru" data-title="Неравенство Птолемея" data-language-autonym="Русский" data-language-local-name="ロシア語" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ptolemajev_izrek" title="スロベニア語: Ptolemajev izrek" lang="sl" hreflang="sl" data-title="Ptolemajev izrek" data-language-autonym="Slovenščina" data-language-local-name="スロベニア語" 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="スウェーデン語: Ptolemaios sats" lang="sv" hreflang="sv" data-title="Ptolemaios sats" data-language-autonym="Svenska" data-language-local-name="スウェーデン語" 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="スワヒリ語: Uhakiki wa Ptolemaio" lang="sw" hreflang="sw" data-title="Uhakiki wa Ptolemaio" data-language-autonym="Kiswahili" data-language-local-name="スワヒリ語" 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="タミル語: தொலெமியின் தேற்றம்" lang="ta" hreflang="ta" data-title="தொலெமியின் தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="タミル語" 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="トルコ語: Batlamyus teoremi" lang="tr" hreflang="tr" data-title="Batlamyus teoremi" data-language-autonym="Türkçe" data-language-local-name="トルコ語" 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="ウクライナ語: Теорема Птолемея" lang="uk" hreflang="uk" data-title="Теорема Птолемея" data-language-autonym="Українська" data-language-local-name="ウクライナ語" 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="ベトナム語: Định lý Ptoleme" lang="vi" hreflang="vi" data-title="Định lý Ptoleme" data-language-autonym="Tiếng Việt" data-language-local-name="ベトナム語" 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="中国語: 托勒密定理" lang="zh" hreflang="zh" data-title="托勒密定理" data-language-autonym="中文" data-language-local-name="中国語" 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="漢文: 托勒密定理" lang="lzh" hreflang="lzh" data-title="托勒密定理" data-language-autonym="文言" data-language-local-name="漢文" class="interlanguage-link-target"><span>文言</span></a></li> </ul> <div class="after-portlet 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title="このページのソースコードを編集する [e]" accesskey="e"><span>編集</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&action=history"><span>履歴を表示</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 全般 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E3%83%AA%E3%83%B3%E3%82%AF%E5%85%83/%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86" title="ここにリンクしている全ウィキページの一覧 [j]" accesskey="j"><span>リンク元</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E9%96%A2%E9%80%A3%E3%83%9A%E3%83%BC%E3%82%B8%E3%81%AE%E6%9B%B4%E6%96%B0%E7%8A%B6%E6%B3%81/%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86" rel="nofollow" title="このページからリンクしているページの最近の更新 [k]" accesskey="k"><span>関連ページの更新状況</span></a></li><li id="t-upload" class="mw-list-item"><a href="//ja.wikipedia.org/wiki/Wikipedia:ファイルのアップロード" title="ファイルをアップロードする [u]" accesskey="u"><span>ファイルをアップロード</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&oldid=103339581" title="このページのこの版への固定リンク"><span>この版への固定リンク</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&action=info" title="このページについての詳細情報"><span>ページ情報</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%81%93%E3%81%AE%E3%83%9A%E3%83%BC%E3%82%B8%E3%82%92%E5%BC%95%E7%94%A8&page=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&id=103339581&wpFormIdentifier=titleform" title="このページの引用方法"><span>このページを引用</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:UrlShortener&url=https%3A%2F%2Fja.wikipedia.org%2Fwiki%2F%25E3%2583%2588%25E3%2583%25AC%25E3%2583%259F%25E3%2583%25BC%25E3%2581%25AE%25E5%25AE%259A%25E7%2590%2586"><span>短縮URLを取得する</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:QrCode&url=https%3A%2F%2Fja.wikipedia.org%2Fwiki%2F%25E3%2583%2588%25E3%2583%25AC%25E3%2583%259F%25E3%2583%25BC%25E3%2581%25AE%25E5%25AE%259A%25E7%2590%2586"><span>QRコードをダウンロード</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> 印刷/書き出し </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%83%96%E3%83%83%E3%82%AF&bookcmd=book_creator&referer=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86"><span>ブックの新規作成</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:DownloadAsPdf&page=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&action=show-download-screen"><span>PDF 形式でダウンロード</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&printable=yes" title="このページの印刷用ページ [p]" accesskey="p"><span>印刷用バージョン</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 他のプロジェクト </div> <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>コモンズ</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="関連付けられたデータリポジトリ項目へのリンク [g]" accesskey="g"><span>ウィキデータ項目</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="ページツール"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="表示"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" 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href="https://jpsearch.go.jp/csearch/jps-cross?csid=jps-cross&keyword=%22%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86%22">ジャパンサーチ</a> <b>·</b> <a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/partners/">TWL</a></span></small></span> <span class="date-container"><i>(<span class="date"><span title="2017年7月">2017年7月</span></span>)</i></span></div></td></tr></tbody></table> <p><b>トレミーの定理</b>（トレミーのていり、<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英</a>: <span lang="en">Ptolemy's Theorem</span>）とは、<a href="/wiki/%E5%86%86_(%E6%95%B0%E5%AD%A6)" title="円 (数学)">円</a>に内接する<a href="/wiki/%E5%9B%9B%E8%A7%92%E5%BD%A2" title="四角形">四角形</a> <i>ABCD</i> において、<a href="/wiki/%E8%BE%BA" title="辺">辺</a>の長さに関する<a href="/wiki/%E7%AD%89%E5%BC%8F" title="等式">等式</a>： </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=AD\cdot BC+AB\cdot DC}"> <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>D</mi> <mo>⋅</mo> <mi>B</mi> <mi>C</mi> <mo>+</mo> <mi>A</mi> <mi>B</mi> <mo>⋅</mo> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC\cdot BD=AD\cdot BC+AB\cdot DC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75a44b3a9561a12e78bbb523049c7a1ad094a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:32.569ex; height:2.343ex;" alt="{\displaystyle AC\cdot BD=AD\cdot BC+AB\cdot DC}"></span></dd></dl> <p>が成り立つという<a href="/wiki/%E3%83%A6%E3%83%BC%E3%82%AF%E3%83%AA%E3%83%83%E3%83%89%E5%B9%BE%E4%BD%95%E5%AD%A6" title="ユークリッド幾何学">幾何学</a>の<a href="/wiki/%E5%AE%9A%E7%90%86" title="定理">定理</a>。トレミーは<a href="/wiki/%E5%8F%A4%E4%BB%A3%E3%83%AD%E3%83%BC%E3%83%9E" title="古代ローマ">古代ローマ</a>の<a href="/wiki/%E5%A4%A9%E6%96%87%E5%AD%A6%E8%80%85" title="天文学者">天文学者</a><a href="/wiki/%E3%82%AF%E3%83%A9%E3%82%A6%E3%83%87%E3%82%A3%E3%82%AA%E3%82%B9%E3%83%BB%E3%83%97%E3%83%88%E3%83%AC%E3%83%9E%E3%82%A4%E3%82%AA%E3%82%B9" title="クラウディオス・プトレマイオス">クラウディオス・プトレマイオス</a>の姓プトレマイオスの英語表記Ptolemyの音訳である。プトレマイオスの定理とも呼ばれる<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>。 </p> <div style="float:right;margin:0 0 1em 1em;"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86.jpg" decoding="async" width="250" height="250" class="mw-file-element" data-file-width="250" data-file-height="250" /></a></span></div> <p>トレミーの定理を一般化した<b>オイラーの定理</b>（オイラーのていり）とは、必ずしも円に内接しない四角形 <i>ABCD</i> において、辺の長さに関する<b>トレミーの不等式</b>（<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英</a>: <span lang="en">Ptolemy's inequality</span>）： </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\leqq AD\cdot BC+AB\cdot DC}"> <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>D</mi> <mo>⋅</mo> <mi>B</mi> <mi>C</mi> <mo>+</mo> <mi>A</mi> <mi>B</mi> <mo>⋅</mo> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC\cdot BD\leqq AD\cdot BC+AB\cdot DC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d985a52f33bd6bca88fc4ccebef756be56473f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.569ex; height:2.676ex;" alt="{\displaystyle AC\cdot BD\leqq AD\cdot BC+AB\cdot DC}"></span></dd></dl> <p>が成り立つという幾何学の定理のことである<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>。逆に、必ずしも同一平面上にない4点 <i>A</i>, <i>B</i>, <i>C</i>, <i>D</i> に関して、辺の長さに関する等式： </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=AD\cdot BC+AB\cdot DC}"> <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>D</mi> <mo>⋅</mo> <mi>B</mi> <mi>C</mi> <mo>+</mo> <mi>A</mi> <mi>B</mi> <mo>⋅</mo> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC\cdot BD=AD\cdot BC+AB\cdot DC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75a44b3a9561a12e78bbb523049c7a1ad094a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:32.569ex; height:2.343ex;" alt="{\displaystyle AC\cdot BD=AD\cdot BC+AB\cdot DC}"></span></dd></dl> <p>が成り立つならば、4点 <i>A</i>, <i>B</i>, <i>C</i>, <i>D</i> は同一直線上にあるか、または同一平面上にあり、かつ四角形 <i>ABCD</i> は同一の円に内接する<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>。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="証明"><span id=".E8.A8.BC.E6.98.8E"></span>証明</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&action=edit&section=1" title="節を編集: 証明"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>計算の便宜をはかり、<i>a</i> = <i>AD</i>, <i>b</i> = <i>AB</i>, <i>c</i> = <i>BC</i>, <i>d</i> = <i>DC</i> とおくことにする。また、<i>A</i> = ∠<i>A</i> = ∠<i>DAB</i>, <i>B</i> = ∠<i>B</i> = ∠<i>ABC</i>, <i>C</i> = ∠<i>C</i> = ∠<i>BCD</i>, <i>D</i> = ∠<i>D</i> = ∠<i>CDA</i> のこととする。 </p><p><a href="/wiki/%E4%BD%99%E5%BC%A6%E5%AE%9A%E7%90%86" title="余弦定理">余弦定理</a>および内接四角形の性質より、 </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 BD^{2}=a^{2}+b^{2}-2ab\cos A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BD^{2}=a^{2}+b^{2}-2ab\cos A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816917d7b00439a05da48d80b2fa42d71a058479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.876ex; height:2.843ex;" alt="{\displaystyle BD^{2}=a^{2}+b^{2}-2ab\cos A}"></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 BD^{2}=c^{2}+d^{2}-2cd\cos C=c^{2}+d^{2}+2cd\cos A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>c</mi> <mi>d</mi> <mi>cos</mi> <mo>⁡</mo> <mi>C</mi> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>c</mi> <mi>d</mi> <mi>cos</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BD^{2}=c^{2}+d^{2}-2cd\cos C=c^{2}+d^{2}+2cd\cos A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bbf87428e565e6ef49bb546109e4522e512d692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:49.018ex; height:2.843ex;" alt="{\displaystyle BD^{2}=c^{2}+d^{2}-2cd\cos C=c^{2}+d^{2}+2cd\cos A}"></span></dd></dl> <p>が成り立つ。ここから cos <i>A</i> を消去して、 </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 (ab+cd)BD^{2}=(ad+bc)(ac+bd)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mi>B</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ab+cd)BD^{2}=(ad+bc)(ac+bd)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb66ccd377e6465ab670766a3ad4463a7cf8c64c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.14ex; height:3.176ex;" alt="{\displaystyle (ab+cd)BD^{2}=(ad+bc)(ac+bd)}"></span></dd></dl> <p>を得る。また <i>AC</i> について同様にして </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 (ad+bc)AC^{2}=(ab+cd)(ac+bd)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mi>A</mi> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ad+bc)AC^{2}=(ab+cd)(ac+bd)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28983ab2787bd4d3c4951ab18764b449153a361f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.993ex; height:3.176ex;" alt="{\displaystyle (ad+bc)AC^{2}=(ab+cd)(ac+bd)}"></span></dd></dl> <p>となるから、2 式を掛けて </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 (ab+cd)(bc+ad)AC^{2}\cdot BD^{2}=(ac+bd)^{2}(ad+bc)(ab+cd)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mi>A</mi> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>B</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ab+cd)(bc+ad)AC^{2}\cdot BD^{2}=(ac+bd)^{2}(ad+bc)(ab+cd)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced9bd490f8633fb89459cb9e2b39c43ca2490f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.668ex; height:3.176ex;" alt="{\displaystyle (ab+cd)(bc+ad)AC^{2}\cdot BD^{2}=(ac+bd)^{2}(ad+bc)(ab+cd)}"></span></dd></dl> <p>を得る。これを整理すれば、 </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=ac+bd}"> <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>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC\cdot BD=ac+bd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f7ec555149e8684ddd1b565609c25ab749e126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.266ex; height:2.343ex;" alt="{\displaystyle AC\cdot BD=ac+bd}"></span></dd></dl> <p>となる。すなわち、 </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=AD\cdot BC+AB\cdot DC}"> <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>D</mi> <mo>⋅</mo> <mi>B</mi> <mi>C</mi> <mo>+</mo> <mi>A</mi> <mi>B</mi> <mo>⋅</mo> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC\cdot BD=AD\cdot BC+AB\cdot DC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75a44b3a9561a12e78bbb523049c7a1ad094a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:32.569ex; height:2.343ex;" alt="{\displaystyle AC\cdot BD=AD\cdot BC+AB\cdot DC}"></span></dd></dl> <p>が示された。 </p> <div class="mw-heading mw-heading3"><h3 id="円に関する反転を用いた証明"><span id=".E5.86.86.E3.81.AB.E9.96.A2.E3.81.99.E3.82.8B.E5.8F.8D.E8.BB.A2.E3.82.92.E7.94.A8.E3.81.84.E3.81.9F.E8.A8.BC.E6.98.8E"></span>円に関する反転を用いた証明</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&action=edit&section=2" title="節を編集: 円に関する反転を用いた証明"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:Ptolemy-crop.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Ptolemy-crop.svg/200px-Ptolemy-crop.svg.png" decoding="async" width="200" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Ptolemy-crop.svg/300px-Ptolemy-crop.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Ptolemy-crop.svg/400px-Ptolemy-crop.svg.png 2x" data-file-width="800" data-file-height="524" /></a><figcaption>円に関する反転を用いた証明</figcaption></figure> <p>Dを中心とする適当な円 <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 \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> に関する<a href="/wiki/%E5%8F%8D%E8%BB%A2%E5%B9%BE%E4%BD%95%E5%AD%A6#円に関する反転" title="反転幾何学">反転</a> によってABCDの外接円が直線に移されるようにする。このとき <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'+B'C'=A'C'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'B'+B'C'=A'C'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b79428294d42ae163cd785d7a04267c5a665870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.657ex; height:2.676ex;" alt="{\displaystyle A'B'+B'C'=A'C'}"></span> が成り立つ。このとき、一般性を失わずに <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 \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> の半径を1と置くことができる。このとき <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',B'C',A'C'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'B',B'C',A'C'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502e648617965688a1e412b9a197f799855a2b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.786ex; height:2.843ex;" alt="{\displaystyle A'B',B'C',A'C'}"></span>はそれぞれ以下のように表される。 </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 {\frac {AB}{DA\cdot DB}},{\frac {BC}{DB\cdot DC}},{\frac {AC}{DA\cdot DC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mrow> <mi>D</mi> <mi>A</mi> <mo>⋅</mo> <mi>D</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mrow> <mi>D</mi> <mi>B</mi> <mo>⋅</mo> <mi>D</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mrow> <mi>D</mi> <mi>A</mi> <mo>⋅</mo> <mi>D</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {AB}{DA\cdot DB}},{\frac {BC}{DB\cdot DC}},{\frac {AC}{DA\cdot DC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace6a91e45011001ba3388803279ef846b34e538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.706ex; height:5.509ex;" alt="{\displaystyle {\frac {AB}{DA\cdot DB}},{\frac {BC}{DB\cdot DC}},{\frac {AC}{DA\cdot DC}}}"></span></dd></dl> <p>この式の両辺に <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 DA\cdot DB\cdot DC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>A</mi> <mo>⋅</mo> <mi>D</mi> <mi>B</mi> <mo>⋅</mo> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DA\cdot DB\cdot DC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c22ddfd2b9a343f3f5dc5c23d22044dc3f6e70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.404ex; height:2.176ex;" alt="{\displaystyle DA\cdot DB\cdot DC}"></span> をかけて最初の式に代入するとトレミーの定理が得られる。 </p> <div class="mw-heading mw-heading2"><h2 id="一般化"><span id=".E4.B8.80.E8.88.AC.E5.8C.96"></span>一般化</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&action=edit&section=3" title="節を編集: 一般化"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一般化に<a href="/wiki/%E3%82%B1%E3%82%A4%E3%82%B7%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86" title="ケイシーの定理">ケイシーの定理</a>がある。 </p> <div class="mw-heading mw-heading2"><h2 id="脚注"><span id=".E8.84.9A.E6.B3.A8"></span>脚注</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&action=edit&section=4" title="節を編集: 脚注"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint" style="float:right; font-size:90%;">[<a href="/wiki/Help:%E8%84%9A%E6%B3%A8/%E8%AA%AD%E8%80%85%E5%90%91%E3%81%91" title="Help:脚注/読者向け"><span title="この欄の操作法">脚注の使い方</span></a>]</div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text"><cite class="citation web" style="font-style:normal" id="CITEREFkotobank-プトレマイオスの定理">デジタル大辞泉. “<a rel="nofollow" class="external text" href="https://kotobank.jp/word/プトレマイオスの定理">プトレマイオスの定理</a>”.  <a href="/wiki/%E3%82%B3%E3%83%88%E3%83%90%E3%83%B3%E3%82%AF" title="コトバンク">コトバンク</a>. <span title="">2019年9月15日</span>閲覧。</cite></span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text"><cite class="citation web" style="font-style:normal" id="CITEREF中村">中村文則. “<a rel="nofollow" class="external text" href="http://izumi-math.jp/F_Nakamura/tolemy/tolemy.pdf">トレミーを散りばめる</a>”.  数学のいずみ. <span title="">2019年9月15日</span>閲覧。</cite></span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"><a href="#CITEREF高木1996">高木 1996</a>, 3 複素数</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86&action=edit&section=5" title="節を編集: 参考文献"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite style="font-style:normal" class="citation book" id="CITEREF高木1996"><a href="/wiki/%E9%AB%98%E6%9C%A8%E8%B2%9E%E6%B2%BB" title="高木貞治">高木貞治</a>『復刻版　近世数学史談・数学雑談』共立出版、1996年12月10日。<style data-mw-deduplicate="TemplateStyles:r101121245">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation.cs-ja1 q,.mw-parser-output .citation.cs-ja2 q{quotes:"「""」""『""』"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.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:var(--color-success,#3a3);margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.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}</style><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%E7%89%B9%E5%88%A5:%E6%96%87%E7%8C%AE%E8%B3%87%E6%96%99/978-4-320-01551-7" title="特別:文献資料/978-4-320-01551-7">978-4-320-01551-7</a>。</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%E5%BE%A9%E5%88%BB%E7%89%88%E3%80%80%E8%BF%91%E4%B8%96%E6%95%B0%E5%AD%A6%E5%8F%B2%E8%AB%87%E3%83%BB%E6%95%B0%E5%AD%A6%E9%9B%91%E8%AB%87&rft.aulast=%E9%AB%98%E6%9C%A8%E8%B2%9E%E6%B2%BB&rft.au=%E9%AB%98%E6%9C%A8%E8%B2%9E%E6%B2%BB&rft.date=1996-12-10&rft.pub=%E5%85%B1%E7%AB%8B%E5%87%BA%E7%89%88&rft.isbn=978-4-320-01551-7&rfr_id=info:sid/ja.wikipedia.org:%E3%83%88%E3%83%AC%E3%83%9F%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="関連項目"><span id=".E9.96.A2.E9.80.A3.E9.A0.85.E7.9B.AE"></span>関連項目</h2><span class="mw-editsection"><span 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