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бакавой панэлі</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">схаваць</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Навігацыя </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage" class="mw-list-item"><a href="/wiki/%D0%93%D0%B0%D0%BB%D0%BE%D1%9E%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D0%B0%D1%80%D0%BE%D0%BD%D0%BA%D0%B0" title="Адкрыць Галоўную старонку [z]" accesskey="z"><span>Галоўная старонка</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%8B%D1%8F:%D0%A1%D1%83%D0%BF%D0%BE%D0%BB%D1%8C%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Пра праект, чым можна заняцца, дзе што шукаць"><span>Супольнасць</span></a></li><li id="n-recentchanges" class="mw-list-item"><a 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[r]" accesskey="r"><span>Апошнія змены</span></a></li><li id="n-newpages" class="mw-list-item"><a href="/wiki/%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:NewPages"><span>Новыя старонкі</span></a></li><li id="n-Village-pump" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%8B%D1%8F:%D0%A4%D0%BE%D1%80%D1%83%D0%BC"><span>Форум</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:Random" title="Паказаць выпадковую старонку [x]" accesskey="x"><span>Выпадковая старонка</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%8B%D1%8F:%D0%94%D0%B0%D0%B2%D0%B5%D0%B4%D0%BA%D0%B0" title="Дзе можна атрымаць тлумачэнні."><span>Даведка</span></a></li><li id="n-bug_in_article" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%8B%D1%8F:%D0%9F%D0%B0%D0%B2%D0%B5%D0%B4%D0%B0%D0%BC%D0%BB%D0%B5%D0%BD%D0%BD%D1%96_%D0%BF%D1%80%D0%B0_%D0%BF%D0%B0%D0%BC%D1%8B%D0%BB%D0%BA%D1%96"><span>Паведаміць пра памылку</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/%D0%93%D0%B0%D0%BB%D0%BE%D1%9E%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D0%B0%D1%80%D0%BE%D0%BD%D0%BA%D0%B0" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Вікіпедыя" src="/static/images/mobile/copyright/wikipedia-wordmark-be.svg" style="width: 7.625em; height: 1.3125em;"> <img class="mw-logo-tagline" alt="" src="/static/images/mobile/copyright/wikipedia-tagline-be.svg" width="117" height="12" style="width: 7.3125em; height: 0.75em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Знайсці ў Вікіпедыі [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Пошук</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Шукаць у Вікіпедыі" aria-label="Шукаць у Вікіпедыі" autocapitalize="sentences" title="Знайсці ў Вікіпедыі [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Адмысловае:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Пошук</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Асабістыя інструменты"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Выгляд"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Выгляд" > <label id="vector-appearance-dropdown-label" for="vector-appearance-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Выгляд</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_be.wikipedia.org&uselang=be" class=""><span>Ахвяраваць</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:CreateAccount&returnto=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F+%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Вам прапануецца стварыць уліковы запіс і ўвайсці ў сістэму, але гэта не абавязкова" class=""><span>Стварыць уліковы запіс</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:UserLogin&returnto=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F+%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Вам прапануецца ўвайсці ў сістэму, але гэта неабавязкова. [o]" accesskey="o" class=""><span>Увайсці</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="Больш магчымасцей" > <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="Асабістыя інструменты" > <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">Асабістыя інструменты</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Меню ўдзельніка" > <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_be.wikipedia.org&uselang=be"><span>Ахвяраваць</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:CreateAccount&returnto=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F+%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Вам прапануецца стварыць уліковы запіс і ўвайсці ў сістэму, але гэта не абавязкова"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Стварыць уліковы запіс</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:UserLogin&returnto=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F+%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Вам прапануецца ўвайсці ў сістэму, але гэта неабавязкова. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Увайсці</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"> Старонкі для ананімных рэдактараў <a href="/wiki/%D0%94%D0%B0%D0%B2%D0%B5%D0%B4%D0%BA%D0%B0:%D0%94%D0%B0%D0%B2%D0%B5%D0%B4%D0%BA%D0%B0" aria-label="Даведацца больш пра праўку"><span>даведацца больш</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/%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:MyContributions" title="Спіс правак, зробленых з гэтага IP-адраса [y]" accesskey="y"><span>Уклад</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:MyTalk" title="Размовы пра праўкі, зробленыя з гэтага IP-адраса [n]" accesskey="n"><span>Размовы</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="Праект"> <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="Змест" 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">Змест</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">перанесці да бакавой панэлі</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">схаваць</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">Пачатак</div> </a> </li> <li id="toc-У_многавугольніку" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#У_многавугольніку"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>У многавугольніку</span> </div> </a> <ul id="toc-У_многавугольніку-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-У_трохвугольніку" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#У_трохвугольніку"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>У трохвугольніку</span> </div> </a> <ul id="toc-У_трохвугольніку-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-У_чатырохвугольніку" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#У_чатырохвугольніку"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>У чатырохвугольніку</span> </div> </a> <ul id="toc-У_чатырохвугольніку-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-У_сферычным_трохвугольніку" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#У_сферычным_трохвугольніку"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>У сферычным трохвугольніку</span> </div> </a> <ul id="toc-У_сферычным_трохвугольніку-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Гл._таксама" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Гл._таксама"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Гл. таксама</span> </div> </a> <ul id="toc-Гл._таксама-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Заўвагі" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Заўвагі"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Заўвагі</span> </div> </a> <ul id="toc-Заўвагі-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Крыніцы" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Крыніцы"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Крыніцы</span> </div> </a> <ul id="toc-Крыніцы-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Літаратура" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Літаратура"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Літаратура</span> </div> </a> <ul id="toc-Літаратура-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="Змест" 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="Пераключыць змест" > <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">Пераключыць змест</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">Упісаная акружнасць</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="Перайсці да артыкула на іншай мове. Даступны на 32 мовах." > <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-32" 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">32 мовы</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/%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9_%D8%AF%D8%A7%D8%AE%D9%84%D9%8A%D8%A9" title="دائرة داخلية – арабская" lang="ar" hreflang="ar" data-title="دائرة داخلية" data-language-autonym="العربية" data-language-local-name="арабская" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A3%D0%BC%D0%B5%D0%B6%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D1%8B%D0%BD%D0%B0" title="Умежаная акружына – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Умежаная акружына" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%B0_%D0%BE%D0%BA%D1%80%D1%8A%D0%B6%D0%BD%D0%BE%D1%81%D1%82" title="Вписана окръжност – балгарская" lang="bg" hreflang="bg" data-title="Вписана окръжност" data-language-autonym="Български" data-language-local-name="балгарская" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%86%D8%A7%D9%88%DB%95_%D8%A8%D8%A7%D8%B2%D9%86%DB%95" title="ناوە بازنە – цэнтральнакурдская" lang="ckb" hreflang="ckb" data-title="ناوە بازنە" data-language-autonym="کوردی" data-language-local-name="цэнтральнакурдская" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kru%C5%BEnice_vepsan%C3%A1" title="Kružnice vepsaná – чэшская" lang="cs" hreflang="cs" data-title="Kružnice vepsaná" data-language-autonym="Čeština" data-language-local-name="чэшская" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A8%D0%B0%D0%BB_%C3%A7%D0%B0%D0%B2%D1%80%D0%B0%D0%BA%C4%83%D1%88" title="Шал çавракăш – чувашская" lang="cv" hreflang="cv" data-title="Шал çавракăш" data-language-autonym="Чӑвашла" data-language-local-name="чувашская" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Indskreven_cirkel" title="Indskreven cirkel – дацкая" lang="da" hreflang="da" data-title="Indskreven cirkel" data-language-autonym="Dansk" data-language-local-name="дацкая" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Inkreis" title="Inkreis – нямецкая" lang="de" hreflang="de" data-title="Inkreis" data-language-autonym="Deutsch" data-language-local-name="нямецкая" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en badge-Q70893996 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Inscribed_circle" title="Inscribed circle – англійская" lang="en" hreflang="en" data-title="Inscribed circle" data-language-autonym="English" data-language-local-name="англійская" 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/Circunferencia_inscrita" title="Circunferencia inscrita – іспанская" lang="es" hreflang="es" data-title="Circunferencia inscrita" data-language-autonym="Español" data-language-local-name="іспанская" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zirkunferentzia_inskribatu" title="Zirkunferentzia inskribatu – баскская" lang="eu" hreflang="eu" data-title="Zirkunferentzia inskribatu" data-language-autonym="Euskara" data-language-local-name="баскская" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa badge-Q70893996 mw-list-item" title=""><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%A7%DB%8C%D8%B1%D9%87_%D9%85%D8%AD%D8%A7%D8%B7%DB%8C_%D9%85%D8%AB%D9%84%D8%AB" title="دایره محاطی مثلث – фарсі" lang="fa" hreflang="fa" data-title="دایره محاطی مثلث" data-language-autonym="فارسی" data-language-local-name="фарсі" 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/Kolmion_sis%C3%A4%C3%A4n_piirretty_ympyr%C3%A4" title="Kolmion sisään piirretty ympyrä – фінская" lang="fi" hreflang="fi" data-title="Kolmion sisään piirretty ympyrä" data-language-autonym="Suomi" data-language-local-name="фінская" 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/Cercle_inscrit" title="Cercle inscrit – французская" lang="fr" hreflang="fr" data-title="Cercle inscrit" data-language-autonym="Français" data-language-local-name="французская" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%92%D7%9C_%D7%97%D7%A1%D7%95%D7%9D" title="מעגל חסום – іўрыт" lang="he" hreflang="he" data-title="מעגל חסום" data-language-autonym="עברית" data-language-local-name="іўрыт" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Be%C3%ADrt_k%C3%B6r_(soksz%C3%B6g)" title="Beírt kör (sokszög) – венгерская" lang="hu" hreflang="hu" data-title="Beírt kör (sokszög)" data-language-autonym="Magyar" data-language-local-name="венгерская" 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%86%D5%A5%D6%80%D5%A3%D5%AE%D5%B5%D5%A1%D5%AC_%D5%B7%D6%80%D5%BB%D5%A1%D5%B6%D5%A1%D5%A3%D5%AB%D5%AE" title="Ներգծյալ շրջանագիծ – армянская" lang="hy" hreflang="hy" data-title="Ներգծյալ շրջանագիծ" data-language-autonym="Հայերեն" data-language-local-name="армянская" 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/Incerchio" title="Incerchio – італьянская" lang="it" hreflang="it" data-title="Incerchio" data-language-autonym="Italiano" data-language-local-name="італьянская" 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/%E5%86%85%E6%8E%A5%E5%86%86" title="内接円 – японская" lang="ja" hreflang="ja" data-title="内接円" data-language-autonym="日本語" data-language-local-name="японская" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%86%E1%9E%93%E1%9E%BB%E1%9E%85%E1%9E%A0%E1%9F%92%E1%9E%82%E1%9F%82%E1%9E%9A%E1%9E%A0%E1%9F%92%E1%9E%82%E1%9F%84%E1%9E%93" title="ចំនុចហ្គែរហ្គោន – кхмерская" lang="km" hreflang="km" data-title="ចំនុចហ្គែរហ្គោន" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="кхмерская" 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/%EB%82%B4%EC%A0%91%EC%9B%90" title="내접원 – карэйская" lang="ko" hreflang="ko" data-title="내접원" data-language-autonym="한국어" data-language-local-name="карэйская" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%BF%D0%B8%D1%88%D0%B0%D0%BD%D0%B0_%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0" title="Впишана кружница – македонская" lang="mk" hreflang="mk" data-title="Впишана кружница" data-language-autonym="Македонски" data-language-local-name="македонская" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Binnenkrink" title="Binnenkrink – ніжненямецкая" lang="nds" hreflang="nds" data-title="Binnenkrink" data-language-autonym="Plattdüütsch" data-language-local-name="ніжненямецкая" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ingeschreven_cirkel" title="Ingeschreven cirkel – нідэрландская" lang="nl" hreflang="nl" data-title="Ingeschreven cirkel" data-language-autonym="Nederlands" data-language-local-name="нідэрландская" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Okr%C4%85g_wpisany" title="Okrąg wpisany – польская" lang="pl" hreflang="pl" data-title="Okrąg wpisany" data-language-autonym="Polski" data-language-local-name="польская" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%BD%D0%B0%D1%8F_%D0%BE%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Вписанная окружность – руская" lang="ru" hreflang="ru" data-title="Вписанная окружность" data-language-autonym="Русский" data-language-local-name="руская" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vp%C3%ADsan%C3%A1_kru%C5%BEnica" title="Vpísaná kružnica – славацкая" lang="sk" hreflang="sk" data-title="Vpísaná kružnica" data-language-autonym="Slovenčina" data-language-local-name="славацкая" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/V%C4%8Drtana_kro%C5%BEnica" title="Včrtana krožnica – славенская" lang="sl" hreflang="sl" data-title="Včrtana krožnica" data-language-autonym="Slovenščina" data-language-local-name="славенская" 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/Inskriven_cirkel" title="Inskriven cirkel – шведская" lang="sv" hreflang="sv" data-title="Inskriven cirkel" data-language-autonym="Svenska" data-language-local-name="шведская" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%B5_%D0%BA%D0%BE%D0%BB%D0%BE" title="Вписане коло – украінская" lang="uk" hreflang="uk" data-title="Вписане коло" data-language-autonym="Українська" data-language-local-name="украінская" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%86%85%E5%88%87%E5%9C%86" title="内切圆 – кітайская" lang="zh" hreflang="zh" data-title="内切圆" data-language-autonym="中文" data-language-local-name="кітайская" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/L%C5%8De-chhiat-%C3%AE%E2%81%BF" title="Lōe-chhiat-îⁿ – Minnan" lang="nan" hreflang="nan" data-title="Lōe-chhiat-îⁿ" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q354337#sitelinks-wikipedia" title="Правіць міжмоўныя спасылкі" class="wbc-editpage">Правіць спасылкі</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Прасторы назваў"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Паказаць уласна змест старонкі [c]" accesskey="c"><span>Старонка</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%D0%A0%D0%B0%D0%B7%D0%BC%D0%BE%D0%B2%D1%8B:%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" rel="discussion" title="Размовы пра змест гэтай старонкі [t]" accesskey="t"><span>Размовы</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Змяніць моўны варыянт" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">беларуская</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Віды"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C"><span>Чытаць</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit" title="Правіць старонку [v]" accesskey="v"><span>Правіць</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit" title="Правіць зыходны код гэтай старонкі [e]" accesskey="e"><span>Правіць зыходнік</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=history" title="Ранейшыя версіі гэтай старонкі. [h]" accesskey="h"><span>Паказаць гісторыю</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Інструменты" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Інструменты</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Інструменты</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">перанесці да бакавой панэлі</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">схаваць</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Больш магчымасцей" > <div class="vector-menu-heading"> Дзеянні </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C"><span>Чытаць</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit" title="Правіць старонку [v]" accesskey="v"><span>Правіць</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit" title="Правіць зыходны код гэтай старонкі [e]" accesskey="e"><span>Правіць зыходнік</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a 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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">З Вікіпедыі, свабоднай энцыклапедыі</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="be" dir="ltr"><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Okrugnost_v_mnogoug.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Okrugnost_v_mnogoug.png/220px-Okrugnost_v_mnogoug.png" decoding="async" width="220" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Okrugnost_v_mnogoug.png/330px-Okrugnost_v_mnogoug.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3a/Okrugnost_v_mnogoug.png 2x" data-file-width="332" data-file-height="385" /></a><figcaption>Акружнасць, упісаная ў многавугольнік ABCDE</figcaption></figure> <p><a href="/wiki/%D0%90%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Акружнасць">Акружнасць</a> называюць <b>упісанаю</b> (<b>умежанаю</b>) у вугал, калі яна ляжыць унутры <a href="/wiki/%D0%92%D1%83%D0%B3%D0%B0%D0%BB" title="Вугал">вугла</a> і датыкаецца да яго старон. Цэнтр акружнасці, упісанай (умежанай) у вугал, ляжыць на <a href="/wiki/%D0%91%D1%96%D1%81%D0%B5%D0%BA%D1%82%D1%80%D1%8B%D1%81%D0%B0" title="Бісектрыса">бісектрысе</a> гэтага вугла. </p><p>Акружнасць называецца <b>ўпісанаю ў выпуклы многавугольнік</b>, калі яна ляжыць унутры дадзенага многавугольніка і датыкаецца да ўсіх яго старон. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="У_многавугольніку"><span id=".D0.A3_.D0.BC.D0.BD.D0.BE.D0.B3.D0.B0.D0.B2.D1.83.D0.B3.D0.BE.D0.BB.D1.8C.D0.BD.D1.96.D0.BA.D1.83"></span>У многавугольніку</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit&section=1" title="Правіць раздзел: У многавугольніку" class="mw-editsection-visualeditor"><span>правіць</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit&section=1" title="Правіць зыходнік раздзела: У многавугольніку"><span>правіць зыходнік</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Калі ў дадзены выпуклы многавугольнік можна ўпісаць (умежыць) акружнасць, то бісектрысы ўсіх унутраных вуглоў дадзенага многавугольніка перасякаюцца ў адным пункце, які з'яўляецца цэнтрам упісанай акружнасці.</li> <li><a href="/wiki/%D0%A0%D0%B0%D0%B4%D1%8B%D1%83%D1%81" title="Радыус">Радыус</a> упісанай у многавугольнік акружнасці роўны адносіне яго плошчы да паўперыметра</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\frac {S}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {S}{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5bb82417b48f0b298d2ee15bee96267935ee715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.482ex; height:5.843ex;" alt="{\displaystyle r={\frac {S}{p}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="У_трохвугольніку"><span id=".D0.A3_.D1.82.D1.80.D0.BE.D1.85.D0.B2.D1.83.D0.B3.D0.BE.D0.BB.D1.8C.D0.BD.D1.96.D0.BA.D1.83"></span>У трохвугольніку</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit&section=2" title="Правіць раздзел: У трохвугольніку" class="mw-editsection-visualeditor"><span>правіць</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit&section=2" title="Правіць зыходнік раздзела: У трохвугольніку"><span>правіць зыходнік</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Уласцівасці ўпісанай акружнасці: </p> <ul><li>У кожны <a href="/wiki/%D0%A2%D1%80%D0%BE%D1%85%D0%B2%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D1%96%D0%BA" title="Трохвугольнік">трохвугольнік</a> можна ўпісаць акружнасць, прытым толькі адну.</li> <li>Цэнтр I упісанай акружнасці называецца <a href="/w/index.php?title=%D0%86%D0%BD%D1%86%D1%8D%D0%BD%D1%82%D1%80&action=edit&redlink=1" class="new" title="Інцэнтр (няма такой старонкі)">інцэнтрам</a>, ён роўнааддалены ад усіх старон і з'яўляецца пунктам перасячэння бісектрыс трохвугольніка.</li> <li>Радыус умежанай у трохвугольнік акружнасці роўны</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\frac {S}{p}}={\sqrt {\frac {(p-a)(p-b)(p-c)}{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mi>p</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {S}{p}}={\sqrt {\frac {(p-a)(p-b)(p-c)}{p}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f8172ccef519b913bc5de8db556842341efc8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.432ex; height:7.509ex;" alt="{\displaystyle r={\frac {S}{p}}={\sqrt {\frac {(p-a)(p-b)(p-c)}{p}}}}"></span></dd></dl> <ul><li>Калі AB — аснова роўнастаронняга <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 ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821677f03b63c3c2e448dffc2ae9c8eea31d9d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.339ex; height:2.176ex;" alt="{\displaystyle \triangle ABC}"></span>, то акружнасць, якая датыкаецца да старон <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> ў пунктах A і B, праходзіць праз інцэнтр трохвугольніка ABC.</li> <li>Формула Эйлера: <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 R^{2}-2Rr=|OI|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>R</mi> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>O</mi> <mi>I</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}-2Rr=|OI|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991d0d900f4b93b7be243a0f174992b4d7a3bbd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.025ex; height:3.343ex;" alt="{\displaystyle R^{2}-2Rr=|OI|^{2}}"></span>, дзе <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> — радыус апісанай вакол трохвугольніка акружнасці, <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> — радыус умежанай у яго акружнасці, O — цэнтр апісанай акружнасці, I — <a href="/w/index.php?title=%D0%86%D0%BD%D1%86%D1%8D%D0%BD%D1%82%D1%80&action=edit&redlink=1" class="new" title="Інцэнтр (няма такой старонкі)">цэнтр упісанай акружнасці</a>.</li> <li>Калі прамая, якая праходзіць праз пункт I паралельная старане AB, перасякае стораны BC і CA у пунктах A<sub>1</sub> і B<sub>1</sub>, то <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}B_{1}=A_{1}B+AB_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>B</mi> <mo>+</mo> <mi>A</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}B_{1}=A_{1}B+AB_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d82dee76ded405209865b10d711b9c24b7b9b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.677ex; height:2.509ex;" alt="{\displaystyle A_{1}B_{1}=A_{1}B+AB_{1}}"></span>.</li> <li>Пункты дотыку ўмежанай у трохвугольнік T акружнасці, злучаныя адрэзкамі, утвараюць трохвугольнік T<sub>1</sub>. <ul><li>Бісектрысы T з'яўляюцца <a href="/wiki/%D0%A1%D1%8F%D1%80%D1%8D%D0%B4%D0%B7%D1%96%D0%BD%D0%BD%D1%8B_%D0%BF%D0%B5%D1%80%D0%BF%D0%B5%D0%BD%D0%B4%D1%8B%D0%BA%D1%83%D0%BB%D1%8F%D1%80" title="Сярэдзінны перпендыкуляр">сярэдзіннымі перпендыкулярамі</a> T<sub>1</sub>.</li> <li>Хай T<sub>2</sub> — ортатрохвугольнік T<sub>1</sub>. Тады яго стораны паралельныя сторанам зыходнага трохвугольніка T.</li> <li>Хай T<sub>3</sub> — пасярэдні трохвугольнік T<sub>1</sub>. Тады бісектрысы T з'яўляюцца вышынямі T<sub>3</sub>.</li> <li>Хай T4 — ортатрохвугольнік T<sub>3</sub>, тады бісектрысы T з'яўляюцца бісектрысамі T<sub>4</sub>.</li></ul></li> <li>Радыус умежанай ў прамавугольны трохвугольнік з катэтамі a, b і гіпатэнузай c акружнасці роўны <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 {a+b-c}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a+b-c}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0acba648bf6b96e690e8f1bf37a1510c137f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.751ex; height:5.343ex;" alt="{\displaystyle {\frac {a+b-c}{2}}}"></span>.</li> <li>Адлегласць ад вяршыні С трохвугольніка да пункта, у якім умежаная акружнасць датыкаецца да стараны, роўная <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {a+b-c}{2}}=p-c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>p</mi> <mo>−</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {a+b-c}{2}}=p-c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f76f1c075b25d68dfb7bb10bf90ce18aff32adc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.181ex; height:5.343ex;" alt="{\displaystyle d={\frac {a+b-c}{2}}=p-c}"></span>.</li> <li>Адлегласць ад вяршыні C да цэнтра ўмежанай акружнасці роўная <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 l_{c}={\frac {r}{\sin({\frac {\gamma }{2}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{c}={\frac {r}{\sin({\frac {\gamma }{2}})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff62312abc45b0bbff174f8430b98347bf769e53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:11.966ex; height:6.176ex;" alt="{\displaystyle l_{c}={\frac {r}{\sin({\frac {\gamma }{2}})}}}"></span>, дзе r — радыус упісанай акружнасці, а γ — вугал вяршыні C.</li> <li>Адлегласць ад вяршыні C да цэнтра ўмежанай акружнасці можна таксама знайсці па формулах <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 l_{c}={\sqrt {(p-c)^{2}+r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{c}={\sqrt {(p-c)^{2}+r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b644eaa02a1d7e753ca0be159dac4bdb79c65464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.883ex; height:4.843ex;" alt="{\displaystyle l_{c}={\sqrt {(p-c)^{2}+r^{2}}}}"></span> і <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 l_{c}={\sqrt {ab-4Rr}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>b</mi> <mo>−</mo> <mn>4</mn> <mi>R</mi> <mi>r</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{c}={\sqrt {ab-4Rr}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0dcac27efc898baf213355b61310916553d9b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.715ex; height:3.176ex;" alt="{\displaystyle l_{c}={\sqrt {ab-4Rr}}}"></span></li> <li>Тэарэма аб трызубцы ці пра канюшыну: Калі <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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> — пункт перасячэння бісектрысы вугла A з апісанаю акружнасцю, а I — цэнтр упісанай акружнасці, то <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 |WI|=|WB|=|WC|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>W</mi> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>W</mi> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>W</mi> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |WI|=|WB|=|WC|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5484cadb423460c538b319812e8db7c152eac27b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.086ex; height:2.843ex;" alt="{\displaystyle |WI|=|WB|=|WC|}"></span>.</li> <li>Лема Вер'ера<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>: хай акружнасць <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> датыкаецца да старон <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> і дугі <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 BC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e0f24a49061dcd63874f7d81f395b5f38800f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\displaystyle BC}"></span> апісанай акружнасці трохвугольніка <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span>. Тады пункты дотыку акружнасці <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> са старонамі і цэнтр умежанай акружнасці трохвугольніка <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> ляжаць на адной прамой.</li></ul> <div class="mw-heading mw-heading2"><h2 id="У_чатырохвугольніку"><span id=".D0.A3_.D1.87.D0.B0.D1.82.D1.8B.D1.80.D0.BE.D1.85.D0.B2.D1.83.D0.B3.D0.BE.D0.BB.D1.8C.D0.BD.D1.96.D0.BA.D1.83"></span>У чатырохвугольніку</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit&section=3" title="Правіць раздзел: У чатырохвугольніку" class="mw-editsection-visualeditor"><span>правіць</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit&section=3" title="Правіць зыходнік раздзела: У чатырохвугольніку"><span>правіць зыходнік</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Апісаны чатырохвугольнік, калі ў яго няма самаперасячэнняў («просты»), павінен быць выпуклым. </p><p>У выпуклы чатырохвугольнік ABCD можна ўпісаць акружнасць тады і толькі тады, калі сумы яго процілеглых старон роўныя: <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 AB+CD=BC+AD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126fc0f308a2102f06e540ef94427cfda97d10eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.175ex; height:2.343ex;" alt="{\displaystyle AB+CD=BC+AD}"></span>. </p><p>Калі ў чатырохвугольнік ўпісана акружнасць, то плошча такога чатырохвугольніка можна вылічыць па формуле: <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 S={\sqrt {(p-a)(p-b)(p-c)(p-d)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\sqrt {(p-a)(p-b)(p-c)(p-d)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374a41c21ab4d7e5fba5e2381c66721570837010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.295ex; height:4.843ex;" alt="{\displaystyle S={\sqrt {(p-a)(p-b)(p-c)(p-d)}}.}"></span> </p><p>Ва ўсякім апісаным чатырохвугольніку сярэдзіны дыяганалей і цэнтр упісанай акружнасці ляжаць на адной прамой (тэарэма Ньютана). На ёй жа ляжыць сярэдзіна адрэзка з канцамі ў пунктах перасячэння процілеглых бакоў чатырохвугольніка. Гэтая прамая называецца прамою Гауса. Цэнтр умежанай у чатырохвугольнік акружнасці — пункт перасячэння вышынь трохвугольніка з вяршынямі ў пункце перасячэння дыяганалей і пунктах скрыжавання процілеглых старон (тэарэма Брокараў). </p> <div class="mw-heading mw-heading2"><h2 id="У_сферычным_трохвугольніку"><span id=".D0.A3_.D1.81.D1.84.D0.B5.D1.80.D1.8B.D1.87.D0.BD.D1.8B.D0.BC_.D1.82.D1.80.D0.BE.D1.85.D0.B2.D1.83.D0.B3.D0.BE.D0.BB.D1.8C.D0.BD.D1.96.D0.BA.D1.83"></span>У сферычным трохвугольніку</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit&section=4" title="Правіць раздзел: У сферычным трохвугольніку" class="mw-editsection-visualeditor"><span>правіць</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit&section=4" title="Правіць зыходнік раздзела: У сферычным трохвугольніку"><span>правіць зыходнік</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Упісаная акружнасць для сферычнага трохвугольніка — гэта акружнасць, якая датыкаецца да ўсіх яго старон. </p> <ul><li>Тангенс радыуса<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>заўв 1<span class="cite-bracket">]</span></a></sup> ўпісанай у сферычны трохвугольнік акружнасці роўны<sup id="cite_ref-St_3-0" class="reference"><a href="#cite_note-St-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference plainlinksneverexpand" style="white-space:nowrap"><sup>:73-74</sup></span></li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tg} r={\sqrt {\frac {\sin(p-a)\sin(p-b)\sin(p-c)}{\sin p}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tg</mi> <mo>⁡</mo> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>p</mi> </mrow> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tg} r={\sqrt {\frac {\sin(p-a)\sin(p-b)\sin(p-c)}{\sin p}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e41525633cec9e1499a504398f91b0eec200753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.181ex; height:7.509ex;" alt="{\displaystyle \operatorname {tg} r={\sqrt {\frac {\sin(p-a)\sin(p-b)\sin(p-c)}{\sin p}}}\,}"></span></dd></dl> <ul><li>Упісаная ў сферычны трохвугольнік акружнасць належыць сферы. Радыус, праведзены з цэнтра сферы праз цэнтр упісанай акружнасці перасячэ сферу ў пункце перасячэння бісектрыс вуглоў (дуг вялікіх колаў сферы, якія дзеляць вуглы напалову) сферычнага трохвугольніка<sup id="cite_ref-St_3-1" class="reference"><a href="#cite_note-St-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference plainlinksneverexpand" style="white-space:nowrap"><sup>:20-21</sup></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Гл._таксама"><span id=".D0.93.D0.BB._.D1.82.D0.B0.D0.BA.D1.81.D0.B0.D0.BC.D0.B0"></span>Гл. таксама</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit&section=5" title="Правіць раздзел: Гл. таксама" class="mw-editsection-visualeditor"><span>правіць</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit&section=5" title="Правіць зыходнік раздзела: Гл. таксама"><span>правіць зыходнік</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=%D0%92%D1%8B%D0%B4%D0%B0%D1%82%D0%BD%D1%8B%D1%8F_%D0%BF%D1%83%D0%BD%D0%BA%D1%82%D1%8B_%D1%82%D1%80%D0%BE%D1%85%D0%B2%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D1%96%D0%BA%D0%B0&action=edit&redlink=1" class="new" title="Выдатныя пункты трохвугольніка (няма такой старонкі)">Выдатныя пункты трохвугольніка</a></li> <li><a href="/wiki/%D0%90%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Апісаная акружнасць">Апісаная акружнасць</a></li> <li><a href="/w/index.php?title=%D0%9F%D0%B0%D0%B7%D0%B0%D1%9E%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit&redlink=1" class="new" title="Пазаўпісаная акружнасць (няма такой старонкі)">Пазаўпісаная акружнасць</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Заўвагі"><span id=".D0.97.D0.B0.D1.9E.D0.B2.D0.B0.D0.B3.D1.96"></span>Заўвагі</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit&section=6" title="Правіць раздзел: Заўвагі" class="mw-editsection-visualeditor"><span>правіць</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit&section=6" title="Правіць зыходнік раздзела: Заўвагі"><span>правіць зыходнік</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Тут радыус акружнасці вымяраецца па сферы, інакш кажучы, гэта градусная мера дугі вялікага круга, якая злучае пункт перасячэння радыуса сферы, праведзенага з цэнтра сферы праз цэнтр акружнасці, са сферай і пункт дотыку акружнасці да стараны трохвугольніка.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Крыніцы"><span id=".D0.9A.D1.80.D1.8B.D0.BD.D1.96.D1.86.D1.8B"></span>Крыніцы</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit&section=7" title="Правіць раздзел: Крыніцы" class="mw-editsection-visualeditor"><span>правіць</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit&section=7" title="Правіць зыходнік раздзела: Крыніцы"><span>правіць зыходнік</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i><span style="white-space: nowrap;">Ефремов Д.</span></i> <a rel="nofollow" class="external text" href="http://ilib.mccme.ru/djvu/ngt/ngt.htm">Новая геометрия треугольника</a>. — Одесса, 1902. — С. 130. — 334 с. <small><span style="white-space: nowrap;"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304132809/http://ilib.mccme.ru/djvu/ngt/ngt.htm">Архівавана</a> 4 сакавіка 2016.</span></small></span></span> </li> <li id="cite_note-St-3"><span class="mw-cite-backlink">↑ <a href="#cite_ref-St_3-0"><sup><i><b>а</b></i></sup></a> <a href="#cite_ref-St_3-1"><sup><i><b>б</b></i></sup></a></span> <span class="reference-text"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i>Степанов Н. Н.</i> Сферическая тригонометрия. — М.—Л.: ОГИЗ, 1948. — 154 с.</span></span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Літаратура"><span id=".D0.9B.D1.96.D1.82.D0.B0.D1.80.D0.B0.D1.82.D1.83.D1.80.D0.B0"></span>Літаратура</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&veaction=edit&section=8" title="Правіць раздзел: Літаратура" class="mw-editsection-visualeditor"><span>правіць</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A3%D0%BF%D1%96%D1%81%D0%B0%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C&action=edit&section=8" title="Правіць зыходнік раздзела: Літаратура"><span>правіць зыходнік</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Факультативный курс по математике. 7-9 / Сост. И. Л. Никольская. — М.: Просвещение, 1991. — С. 89. — 383 с. — <a href="/wiki/%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:BookSources/5090012873" class="internal mw-magiclink-isbn">ISBN 5-09-001287-3</a></li> <li><i>Понарин Я. П.</i> Элементарная геометрия. В 2 тт. — М.: МЦНМО, 2004. — С. 52-53. — <a href="/wiki/%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:BookSources/5940571700" class="internal mw-magiclink-isbn">ISBN 5-94057-170-0</a></li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Узята з "<a dir="ltr" href="https://be.wikipedia.org/w/index.php?title=Упісаная_акружнасць&oldid=4845078">https://be.wikipedia.org/w/index.php?title=Упісаная_акружнасць&oldid=4845078</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%90%D0%B4%D0%BC%D1%8B%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D0%B5:Categories" title="Адмысловае:Categories">Катэгорыі</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D1%8D%D0%B3%D0%BE%D1%80%D1%8B%D1%8F:%D0%93%D0%B5%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%8F_%D1%82%D1%80%D0%BE%D1%85%D0%B2%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D1%96%D0%BA%D0%B0" title="Катэгорыя:Геаметрыя трохвугольніка">Геаметрыя трохвугольніка</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D1%8D%D0%B3%D0%BE%D1%80%D1%8B%D1%8F:%D0%9A%D1%80%D1%8B%D0%B2%D1%8B%D1%8F" title="Катэгорыя:Крывыя">Крывыя</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Схаваныя катэгорыі: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D1%8D%D0%B3%D0%BE%D1%80%D1%8B%D1%8F:%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%8B%D1%8F:%D0%A8%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD_%C2%AB%D0%92%D0%BE%D0%BD%D0%BA%D0%B0%D0%B2%D1%8B%D1%8F_%D1%81%D0%BF%D0%B0%D1%81%D1%8B%D0%BB%D0%BA%D1%96%C2%BB_%D0%BF%D1%83%D1%81%D1%82%D1%8B" title="Катэгорыя:Вікіпедыя:Шаблон «Вонкавыя спасылкі» пусты">Вікіпедыя:Шаблон «Вонкавыя спасылкі» пусты</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D1%8D%D0%B3%D0%BE%D1%80%D1%8B%D1%8F:%D0%A1%D1%82%D0%B0%D1%80%D0%BE%D0%BD%D0%BA%D1%96,_%D0%BD%D0%B0_%D1%8F%D0%BA%D1%96%D1%85_%D1%83%D0%B6%D1%8B%D1%82%D1%8B_%D1%87%D0%B0%D1%80%D0%BE%D1%9E%D0%BD%D1%8B%D1%8F_%D1%81%D0%BF%D0%B0%D1%81%D1%8B%D0%BB%D0%BA%D1%96_ISBN" title="Катэгорыя:Старонкі, на якіх ужыты чароўныя спасылкі ISBN">Старонкі, на якіх ужыты чароўныя спасылкі ISBN</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"> Апошняе змяненне старонкі адбылося 13:04, 24 верасня 2024.</li> <li id="footer-info-copyright">Тэкст даступны на ўмовах <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.be">ліцэнзіі Creative Commons Attribution-ShareAlike</a>, у асобных выпадках могуць дзейнічаць дадатковыя ўмовы. 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