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[o]" accesskey="o" class=""><span>Ieiet Vikipēdijā</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Vairāk iespēju" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Dalībnieka rīki" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Dalībnieka rīki</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Dalībnieka izvēlne" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_lv.wikipedia.org&uselang=lv"><span>Ziedojumi</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Ptolemaja+teor%C4%93ma" title="Ieteicams izveidot kontu un pieslēgties; tomēr tas nav obligāti."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Izveidot jaunu Vikipēdijas kontu</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Ptolemaja+teor%C4%93ma" title="Aicinām tevi ieiet Vikipēdijā, tomēr tas nav obligāti. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Ieiet Vikipēdijā</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"> Lapas redaktoriem, kuri nav pieslēgušies <a href="/wiki/Vikip%C4%93dija:Ievads" aria-label="Uzzini vairāk par rediģēšanu"><span>uzzināt vairāk</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/Special:MyContributions" title="Labojumi, kas veikti no šīs IP adreses [y]" accesskey="y"><span>Devums</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Diskusija par labojumiem, kas izdarīti no šīs IP adreses [n]" accesskey="n"><span>Diskusijas</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="Vietne"> <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="Saturs" 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">Saturs</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pārvietot uz sānjoslu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">paslēpt</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">Sākums</div> </a> </li> <li id="toc-Pierādījums_ar_līdzīgiem_trijstūriem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pierādījums_ar_līdzīgiem_trijstūriem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Pierādījums ar līdzīgiem trijstūriem</span> </div> </a> <ul id="toc-Pierādījums_ar_līdzīgiem_trijstūriem-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="Saturs" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="Pārslēgt satura rādītāju" > <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">Pārslēgt satura rādītāju</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">Ptolemaja teorēma</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="Pāriet uz rakstu citā valodā. Pieejams 38 valodās" > <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 valodas</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="مبرهنة بطليموس – arābu" lang="ar" hreflang="ar" data-title="مبرهنة بطليموس" data-language-autonym="العربية" data-language-local-name="arābu" 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 – katalāņu" lang="ca" hreflang="ca" data-title="Teorema de Ptolemeu" data-language-autonym="Català" data-language-local-name="katalāņu" 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 – centrālkurdu" lang="ckb" hreflang="ckb" data-title="تیۆرمی بەتڵیمۆس" data-language-autonym="کوردی" data-language-local-name="centrālkurdu" 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 – vācu" lang="de" hreflang="de" data-title="Satz von Ptolemäus" data-language-autonym="Deutsch" data-language-local-name="vācu" 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 – grieķu" lang="el" hreflang="el" data-title="Θεώρημα του Πτολεμαίου" data-language-autonym="Ελληνικά" data-language-local-name="grieķu" 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 – angļu" lang="en" hreflang="en" data-title="Ptolemy's theorem" data-language-autonym="English" data-language-local-name="angļu" 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 – spāņu" lang="es" hreflang="es" data-title="Teorema de Ptolomeo" data-language-autonym="Español" data-language-local-name="spāņu" 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 – persiešu" lang="fa" hreflang="fa" data-title="قضیه بطلمیوس" data-language-autonym="فارسی" data-language-local-name="persiešu" 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 – somu" lang="fi" hreflang="fi" data-title="Ptolemaioksen lause" data-language-autonym="Suomi" data-language-local-name="somu" 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 – franču" lang="fr" hreflang="fr" data-title="Théorème de Ptolémée" data-language-autonym="Français" data-language-local-name="franču" 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 – galisiešu" lang="gl" hreflang="gl" data-title="Teorema de Tolomeo" data-language-autonym="Galego" data-language-local-name="galisiešu" 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 – ivrits" lang="he" hreflang="he" data-title="משפט תלמי" data-language-autonym="עברית" data-language-local-name="ivrits" 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 – hindi" lang="hi" hreflang="hi" data-title="प्तोलेमी का प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="hindi" 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 – horvātu" lang="hr" hreflang="hr" data-title="Ptolemejev poučak" data-language-autonym="Hrvatski" data-language-local-name="horvātu" 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 – ungāru" lang="hu" hreflang="hu" data-title="Ptolemaiosz-tétel" data-language-autonym="Magyar" data-language-local-name="ungāru" 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 – armēņu" lang="hy" hreflang="hy" data-title="Պտղոմեոսի թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="armēņu" 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 – itāļu" lang="it" hreflang="it" data-title="Teorema di Tolomeo" data-language-autonym="Italiano" data-language-local-name="itāļu" 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 – japāņu" lang="ja" hreflang="ja" data-title="トレミーの定理" data-language-autonym="日本語" data-language-local-name="japāņu" 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 – gruzīnu" lang="ka" hreflang="ka" data-title="პტოლემეს თეორემა" data-language-autonym="ქართული" data-language-local-name="gruzīnu" 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 – khmeru" lang="km" hreflang="km" data-title="ទ្រឹស្តីបទតូលេមី" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmeru" 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 – korejiešu" lang="ko" hreflang="ko" data-title="프톨레마이오스 정리" data-language-autonym="한국어" data-language-local-name="korejiešu" 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 – lietuviešu" lang="lt" hreflang="lt" data-title="Ptolemėjaus teorema" data-language-autonym="Lietuvių" data-language-local-name="lietuviešu" class="interlanguage-link-target"><span>Lietuvių</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 – maķedoniešu" lang="mk" hreflang="mk" data-title="Птоломеева теорема" data-language-autonym="Македонски" data-language-local-name="maķedoniešu" 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 – holandiešu" lang="nl" hreflang="nl" data-title="Stelling van Ptolemaeus" data-language-autonym="Nederlands" data-language-local-name="holandiešu" 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 – jaunnorvēģu" lang="nn" hreflang="nn" data-title="Ptolemaios-satsen" data-language-autonym="Norsk nynorsk" data-language-local-name="jaunnorvēģu" 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 – poļu" lang="pl" hreflang="pl" data-title="Twierdzenie Ptolemeusza" data-language-autonym="Polski" data-language-local-name="poļu" 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 – portugāļu" lang="pt" hreflang="pt" data-title="Teorema de Ptolomeu" data-language-autonym="Português" data-language-local-name="portugāļu" 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 – rumāņu" lang="ro" hreflang="ro" data-title="Teorema lui Ptolemeu" data-language-autonym="Română" data-language-local-name="rumāņu" 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="Неравенство Птолемея – krievu" lang="ru" hreflang="ru" data-title="Неравенство Птолемея" data-language-autonym="Русский" data-language-local-name="krievu" 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="Ptolemajev izrek – slovēņu" lang="sl" hreflang="sl" data-title="Ptolemajev izrek" data-language-autonym="Slovenščina" data-language-local-name="slovēņu" 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 – zviedru" lang="sv" hreflang="sv" data-title="Ptolemaios sats" data-language-autonym="Svenska" data-language-local-name="zviedru" 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 – svahili" lang="sw" hreflang="sw" data-title="Uhakiki wa Ptolemaio" data-language-autonym="Kiswahili" data-language-local-name="svahili" 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 – tamilu" lang="ta" hreflang="ta" data-title="தொலெமியின் தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="tamilu" 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 – turku" lang="tr" hreflang="tr" data-title="Batlamyus teoremi" data-language-autonym="Türkçe" data-language-local-name="turku" 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 – ukraiņu" lang="uk" hreflang="uk" data-title="Теорема Птолемея" data-language-autonym="Українська" data-language-local-name="ukraiņu" 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" 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data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Izskats</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">pārvietot uz sānjoslu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">paslēpt</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Vikipēdijas lapa</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="lv" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Att%C4%93ls:Ptolemy_equality.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Ptolemy_equality.svg/300px-Ptolemy_equality.svg.png" decoding="async" width="300" height="281" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Ptolemy_equality.svg/450px-Ptolemy_equality.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Ptolemy_equality.svg/600px-Ptolemy_equality.svg.png 2x" data-file-width="200" data-file-height="187" /></a><figcaption>Ptolemaja teorēma ir sakarība starp ievilkta četrstūra garumiem.<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 \definecolor {V}{rgb}{0.5803921568627451,0,0.8274509803921568}\definecolor {B}{rgb}{0,0,1}\definecolor {R}{rgb}{0.8,0,0}{\color {V}AC}\cdot {\color {V}BD}={\color {B}AB}\cdot {\color {B}CD}+{\color {R}BC}\cdot {\color {R}AD}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#9400d3"> <mi>A</mi> <mi>C</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#9400d3"> <mi>B</mi> <mi>D</mi> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#cc0000"> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#cc0000"> <mi>A</mi> <mi>D</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \definecolor {V}{rgb}{0.5803921568627451,0,0.8274509803921568}\definecolor {B}{rgb}{0,0,1}\definecolor {R}{rgb}{0.8,0,0}{\color {V}AC}\cdot {\color {V}BD}={\color {B}AB}\cdot {\color {B}CD}+{\color {R}BC}\cdot {\color {R}AD}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5175aa8b7afe40477a189ed103cd095b397359" 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 \definecolor {V}{rgb}{0.5803921568627451,0,0.8274509803921568}\definecolor {B}{rgb}{0,0,1}\definecolor {R}{rgb}{0.8,0,0}{\color {V}AC}\cdot {\color {V}BD}={\color {B}AB}\cdot {\color {B}CD}+{\color {R}BC}\cdot {\color {R}AD}}"></span></figcaption></figure> <p><b>Ptolemaja teorēma</b> ir sakarība <a href="/wiki/Ri%C5%86%C4%B7a_l%C4%ABnija#Ievilktas_riņķa_līnijas" title="Riņķa līnija">ievilktā četrstūrī</a> starp tā malām un <a href="/wiki/Diagon%C4%81le" title="Diagonāle">diagonālēm</a>. <a href="/wiki/Ptolemajs" title="Ptolemajs">Ptolemajs</a> to izmantoja, lai aprēķinātu <a href="/wiki/Horda_(matem%C4%81tika)" title="Horda (matemātika)">hordu</a> garumus dažādiem <a href="/wiki/Le%C5%86%C4%B7is" title="Leņķis">leņķiem</a> no kā var iegūt, piemēram, <a href="/wiki/Sinuss" title="Sinuss">sinusa</a> vērtību tabulu. Ja ievilktā četrstūra virsotnes ir secīgi <i>A, B, C, D</i>, tad teorēma apgalvo, ka <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=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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC\cdot BD=AB\cdot CD+BC\cdot AD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a3355d706c67c614eed4dc8362f492cb1c6923" 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=AB\cdot CD+BC\cdot AD}"></span>. </p><p>Ar Ptolemaja teorēmu var pierādīt dažādas citas teorēmas, piemēram, <a href="/wiki/Pitagora_teor%C4%93ma" title="Pitagora teorēma">Pitagora teorēmu</a>, <a href="/wiki/Kosinusu_teor%C4%93ma" title="Kosinusu teorēma">kosinusu teorēmu</a>, sinusa <a href="/wiki/Sinuss#Sakarības" title="Sinuss">leņķu summu</a>, sinusa leņķu starpību, <a href="/wiki/Kosinuss" title="Kosinuss">kosinusa</a> leņķu summu. </p> <div class="mw-heading mw-heading2"><h2 id="Pierādījums_ar_līdzīgiem_trijstūriem"><span id="Pier.C4.81d.C4.ABjums_ar_l.C4.ABdz.C4.ABgiem_trijst.C5.ABriem"></span>Pierādījums ar līdzīgiem trijstūriem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ptolemaja_teor%C4%93ma&veaction=edit&section=1" title="Labot sadaļu: Pierādījums ar līdzīgiem trijstūriem" class="mw-editsection-visualeditor"><span>labot šo sadaļu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ptolemaja_teor%C4%93ma&action=edit&section=1" title="Labot sadaļas vikikodu: Pierādījums ar līdzīgiem trijstūriem"><span>labot pirmkodu</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Att%C4%93ls:Ptolemy%27s_theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Ptolemy%27s_theorem.svg/300px-Ptolemy%27s_theorem.svg.png" decoding="async" width="300" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Ptolemy%27s_theorem.svg/450px-Ptolemy%27s_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Ptolemy%27s_theorem.svg/600px-Ptolemy%27s_theorem.svg.png 2x" data-file-width="500" data-file-height="164" /></a><figcaption>1) Tiek atrasti vienādie leņķi (<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 BAC=\angle BDC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle BAC=\angle BDC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8497aa1bf16bd680a6cd82cace0622c7fa27c8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.183ex; height:2.176ex;" alt="{\displaystyle \angle BAC=\angle BDC}"></span>un <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 ACB=\angle ADB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>D</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ACB=\angle ADB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0cbb5f229b02103d8edbe8af8d70edd08cca8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.159ex; height:2.176ex;" alt="{\displaystyle \angle ACB=\angle ADB}"></span>) kā arī definēti leņķi (<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 CBD=\angle ABK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>D</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle CBD=\angle ABK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c57efec743819ed046c6d379063e37a830e4fe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.482ex; height:2.176ex;" alt="{\displaystyle \angle CBD=\angle ABK}"></span>), no kā izriet <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 CBK=\angle ABD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>K</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle CBK=\angle ABD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36ef706068490217bcdc45810691195b179442ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.482ex; height:2.176ex;" alt="{\displaystyle \angle CBK=\angle ABD}"></span>. 2) Tiek atrasti līdzīgie trijstūri <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 \triangle ABK\sim \triangle BCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>K</mi> <mo>∼</mo> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABK\sim \triangle BCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0dfaea1fe7b8565dc212af23d9b2c1bd8a85851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.258ex; height:2.176ex;" alt="{\displaystyle \triangle ABK\sim \triangle BCD}"></span> un <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 \triangle ABD\sim \triangle CBK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>D</mi> <mo>∼</mo> <mi mathvariant="normal">△</mi> <mi>C</mi> <mi>B</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABD\sim \triangle CBK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1abae942f3dc0e8d5b31880427e9b9f971fd5c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.258ex; height:2.176ex;" alt="{\displaystyle \triangle ABD\sim \triangle CBK}"></span>.3) Tiek iegūtas līdzīgo trijstūru malu attiecības <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 {AK}{AB}}={\frac {CD}{BD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>K</mi> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>C</mi> <mi>D</mi> </mrow> <mrow> <mi>B</mi> <mi>D</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {AK}{AB}}={\frac {CD}{BD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a71dc358968886aaef9c0268c3a72eed3723e4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.27ex; height:5.509ex;" alt="{\displaystyle {\frac {AK}{AB}}={\frac {CD}{BD}}}"></span> un <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 {AD}{BD}}={\frac {CK}{BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>D</mi> </mrow> <mrow> <mi>B</mi> <mi>D</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>C</mi> <mi>K</mi> </mrow> <mrow> <mi>B</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {AD}{BD}}={\frac {CK}{BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea1f8a108dd9479bcc83bddc1697f43a3299187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.291ex; height:5.509ex;" alt="{\displaystyle {\frac {AD}{BD}}={\frac {CK}{BC}}}"></span> un tās summējot iegūst meklēto <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\cdot AC=CD\cdot AB+AD\cdot BC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>D</mi> <mo>⋅</mo> <mi>A</mi> <mi>C</mi> <mo>=</mo> <mi>C</mi> <mi>D</mi> <mo>⋅</mo> <mi>A</mi> <mi>B</mi> <mo>+</mo> <mi>A</mi> <mi>D</mi> <mo>⋅</mo> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BD\cdot AC=CD\cdot AB+AD\cdot BC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b95781470c01ced0845240dafd61428f681cbf3b" 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 BD\cdot AC=CD\cdot AB+AD\cdot BC}"></span>.</figcaption></figure> <p>Ievilktam četrstūrim ABCD <a href="/wiki/Ievilkts_le%C5%86%C4%B7is" title="Ievilkts leņķis">ievilktie leņķi</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 \angle BAC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle BAC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620f242c2371936d9c9bd90d230623d202f42e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.952ex; height:2.176ex;" alt="{\displaystyle \angle BAC}"></span> un <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 BDC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle BDC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd3a970aa5ea1c53f7dbb17a8332ca477073d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.133ex; height:2.176ex;" alt="{\displaystyle \angle BDC}"></span> attiecas uz vienu <a href="/wiki/Loks_(matem%C4%81tika)" title="Loks (matemātika)">loku</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 {\overset {\frown }{BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d5cffe92c89482266f356039a49ba3d4333277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{BC}}}"></span>, tādēļ tie ir vienādi. Pēc tās pašas argumentācijas ievilktie leņķi <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 ACB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ACB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cb3ffaff70a9b25c9564d8ef5d9f801222754c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.952ex; height:2.176ex;" alt="{\displaystyle \angle ACB}"></span> un <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 ADB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>D</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ADB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6172339b15441629b149f0e6ca7b96c6be6e4d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.109ex; height:2.176ex;" alt="{\displaystyle \angle ADB}"></span> attiecas uz loku <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 {\overset {\frown }{AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>⌢</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {\frown }{AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4608fbf7084ad30e15487e824daf74260640f810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:3.343ex;" alt="{\displaystyle {\overset {\frown }{AB}}}"></span>, tādēļ tie ir vienādi. Var ieviest tādu punktu <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> uz līnijas <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b930d133ca536a071bec52a9acc4b05482890d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.509ex; height:2.176ex;" alt="{\displaystyle AC}"></span>, ka izpildās <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 CBD=\angle ABK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>D</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle CBD=\angle ABK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c57efec743819ed046c6d379063e37a830e4fe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.482ex; height:2.176ex;" alt="{\displaystyle \angle CBD=\angle ABK}"></span>. No tā var secināt, ka <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 ABK+\angle CBK=\angle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>K</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>K</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ABK+\angle CBK=\angle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8500f59f158d56526a2b1cfcac4661571f932243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:27.416ex; height:2.343ex;" alt="{\displaystyle \angle ABK+\angle CBK=\angle ABC}"></span> un <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 ABD+\angle CBD=\angle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>D</mi> <mo>+</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>D</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ABD+\angle CBD=\angle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23f91c966d6a3976fb220bb47332829c773c288c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:27.132ex; height:2.343ex;" alt="{\displaystyle \angle ABD+\angle CBD=\angle ABC}"></span>, tad pielīdzinot un atceroties, ka <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 ABK=\angle CBD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>K</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ABK=\angle CBD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24373a1b30393e5cf57f194b7b486c67803aa8b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.482ex; height:2.176ex;" alt="{\displaystyle \angle ABK=\angle CBD}"></span> iegūst <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 CBK=\angle ABD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>K</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle CBK=\angle ABD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36ef706068490217bcdc45810691195b179442ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.482ex; height:2.176ex;" alt="{\displaystyle \angle CBK=\angle ABD}"></span>. </p><p>Tagad <a href="/wiki/Trijst%C5%ABris" title="Trijstūris">trijstūriem</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 \triangle ABK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6ec33cce7723f0974d7594b3a314356a7d473b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.639ex; height:2.176ex;" alt="{\displaystyle \triangle ABK}"></span> un <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 \triangle BCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle BCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc576c465970dba4eb550de58109e07508f2552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.521ex; height:2.176ex;" alt="{\displaystyle \triangle BCD}"></span> ir divi vienādi leņķi (<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 BAK=\angle BDC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>A</mi> <mi>K</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle BAK=\angle BDC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5f27549911625f415fada1ebcab9433033b768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.482ex; height:2.176ex;" alt="{\displaystyle \angle BAK=\angle BDC}"></span> un <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 ABK=\angle CBD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>K</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ABK=\angle CBD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24373a1b30393e5cf57f194b7b486c67803aa8b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.482ex; height:2.176ex;" alt="{\displaystyle \angle ABK=\angle CBD}"></span>), tādēļ tie ir <a href="/w/index.php?title=L%C4%ABdz%C4%ABgi_trijst%C5%ABri&action=edit&redlink=1" class="new" title="Līdzīgi trijstūri (vēl nav uzrakstīts)">līdzīgi trijstūri</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 \triangle ABK\sim \triangle BCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>K</mi> <mo>∼</mo> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABK\sim \triangle BCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0dfaea1fe7b8565dc212af23d9b2c1bd8a85851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.258ex; height:2.176ex;" alt="{\displaystyle \triangle ABK\sim \triangle BCD}"></span>. Pēc tās pašas argumentācijas trijstūriem <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 \triangle ABD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06a58977c380ed00aa14f2d4a5a885bb5b220769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.497ex; height:2.176ex;" alt="{\displaystyle \triangle ABD}"></span> un <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 \triangle CBK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>C</mi> <mi>B</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle CBK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df47912e881d01b8e8a5bc6541153aebf2c9b2d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.662ex; height:2.176ex;" alt="{\displaystyle \triangle CBK}"></span> ir divi kopīgi leņķi (<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 ABD=\angle CBK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>B</mi> <mi>D</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>C</mi> <mi>B</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ABD=\angle CBK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f55615ddee814862e1449b10e13a7a91566b074e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.482ex; height:2.176ex;" alt="{\displaystyle \angle ABD=\angle CBK}"></span> un <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 ADB=\angle BCK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mi>A</mi> <mi>D</mi> <mi>B</mi> <mo>=</mo> <mi mathvariant="normal">∠</mi> <mi>B</mi> <mi>C</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle ADB=\angle BCK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4e91a6b271e2f3f22d789cc0cb47e8b2cda8a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.482ex; height:2.176ex;" alt="{\displaystyle \angle ADB=\angle BCK}"></span>), tādēļ tie ir līdzīgi trijstūri <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 \triangle ABD\sim \triangle CBK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>D</mi> <mo>∼</mo> <mi mathvariant="normal">△</mi> <mi>C</mi> <mi>B</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABD\sim \triangle CBK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1abae942f3dc0e8d5b31880427e9b9f971fd5c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.258ex; height:2.176ex;" alt="{\displaystyle \triangle ABD\sim \triangle CBK}"></span>. </p><p>Līdzīgi trijstūri saglabā malu attiecības, tādēļ no līdzīgiem trijstūriem <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 \triangle ABK\sim \triangle BCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>K</mi> <mo>∼</mo> <mi mathvariant="normal">△</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABK\sim \triangle BCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0dfaea1fe7b8565dc212af23d9b2c1bd8a85851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.258ex; height:2.176ex;" alt="{\displaystyle \triangle ABK\sim \triangle BCD}"></span> iegūst <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 {AK}{AB}}={\frac {CD}{BD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>K</mi> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>C</mi> <mi>D</mi> </mrow> <mrow> <mi>B</mi> <mi>D</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {AK}{AB}}={\frac {CD}{BD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a71dc358968886aaef9c0268c3a72eed3723e4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.27ex; height:5.509ex;" alt="{\displaystyle {\frac {AK}{AB}}={\frac {CD}{BD}}}"></span> un no līdzīgiem trijstūriem <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 \triangle ABD\sim \triangle CBK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>D</mi> <mo>∼</mo> <mi mathvariant="normal">△</mi> <mi>C</mi> <mi>B</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABD\sim \triangle CBK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1abae942f3dc0e8d5b31880427e9b9f971fd5c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.258ex; height:2.176ex;" alt="{\displaystyle \triangle ABD\sim \triangle CBK}"></span> iegūst <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 {AD}{BD}}={\frac {CK}{BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>D</mi> </mrow> <mrow> <mi>B</mi> <mi>D</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>C</mi> <mi>K</mi> </mrow> <mrow> <mi>B</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {AD}{BD}}={\frac {CK}{BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea1f8a108dd9479bcc83bddc1697f43a3299187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.291ex; height:5.509ex;" alt="{\displaystyle {\frac {AD}{BD}}={\frac {CK}{BC}}}"></span>. Katrā izteiksmē pareiznot ar saucējiem iegūst: <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 AK\cdot BD=CD\cdot AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>K</mi> <mo>⋅</mo> <mi>B</mi> <mi>D</mi> <mo>=</mo> <mi>C</mi> <mi>D</mi> <mo>⋅</mo> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AK\cdot BD=CD\cdot AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04f04892a23a48f011b9059e939866bef86cd5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.152ex; height:2.176ex;" alt="{\displaystyle AK\cdot BD=CD\cdot AB}"></span> un <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\cdot BC=CK\cdot BD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>D</mi> <mo>⋅</mo> <mi>B</mi> <mi>C</mi> <mo>=</mo> <mi>C</mi> <mi>K</mi> <mo>⋅</mo> <mi>B</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AD\cdot BC=CK\cdot BD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a8ed86bde7dfbceb367ed32bfe5a8d286af390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.175ex; height:2.176ex;" alt="{\displaystyle AD\cdot BC=CK\cdot BD}"></span>. <a href="/wiki/Summa" title="Summa">Summējot</a> abas izteiksmes iegūst: <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 AK\cdot BD+CK\cdot BD=CD\cdot AB+AD\cdot BC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>K</mi> <mo>⋅</mo> <mi>B</mi> <mi>D</mi> <mo>+</mo> <mi>C</mi> <mi>K</mi> <mo>⋅</mo> <mi>B</mi> <mi>D</mi> <mo>=</mo> <mi>C</mi> <mi>D</mi> <mo>⋅</mo> <mi>A</mi> <mi>B</mi> <mo>+</mo> <mi>A</mi> <mi>D</mi> <mo>⋅</mo> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AK\cdot BD+CK\cdot BD=CD\cdot AB+AD\cdot BC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92ba9bce971b045825532770f9d81da9525d076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:44.909ex; height:2.343ex;" alt="{\displaystyle AK\cdot BD+CK\cdot BD=CD\cdot AB+AD\cdot BC}"></span>, iznesot kopējo reizinātāju: <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(AK+CK)=CD\cdot AB+AD\cdot BC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>D</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mi>K</mi> <mo>+</mo> <mi>C</mi> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mi>D</mi> <mo>⋅</mo> <mi>A</mi> <mi>B</mi> <mo>+</mo> <mi>A</mi> <mi>D</mi> <mo>⋅</mo> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BD(AK+CK)=CD\cdot AB+AD\cdot BC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05167b753189a1965a512c0a5af1025bcb3ce3e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.672ex; height:2.843ex;" alt="{\displaystyle BD(AK+CK)=CD\cdot AB+AD\cdot BC}"></span>, savukārt <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 AK+CK=AC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>K</mi> <mo>+</mo> <mi>C</mi> <mi>K</mi> <mo>=</mo> <mi>A</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AK+CK=AC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cca31cc0977f9275815edc2eb99360794307f258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.09ex; height:2.343ex;" alt="{\displaystyle AK+CK=AC}"></span>, tādēļ ievietojot iegūst <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\cdot AC=CD\cdot AB+AD\cdot BC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>D</mi> <mo>⋅</mo> <mi>A</mi> <mi>C</mi> <mo>=</mo> <mi>C</mi> <mi>D</mi> <mo>⋅</mo> <mi>A</mi> <mi>B</mi> <mo>+</mo> <mi>A</mi> <mi>D</mi> <mo>⋅</mo> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BD\cdot AC=CD\cdot AB+AD\cdot BC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b95781470c01ced0845240dafd61428f681cbf3b" 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 BD\cdot AC=CD\cdot AB+AD\cdot BC}"></span> <a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a> </p><p>Šis pierādījums izpildās tikai, ja četrstūra malas viena otru nešķērso, taču teorēma ir spēkā arī šķērsojošu malu gadījumā. </p>    </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="">Saturs iegūts no "<a dir="ltr" href="https://lv.wikipedia.org/w/index.php?title=Ptolemaja_teorēma&oldid=4111663">https://lv.wikipedia.org/w/index.php?title=Ptolemaja_teorēma&oldid=4111663</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categories" title="Special:Categories">Kategorija</a>: <ul><li><a href="/wiki/Kategorija:%C4%A2eometrija" title="Kategorija:Ģeometrija">Ģeometrija</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Šī lapa pēdējoreiz labota: 2024. gada 22. augusts plkst. 09.17.</li> <li id="footer-info-copyright">Teksts ir pieejams saskaņā ar <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution/Share-Alike licenci</a>; 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