Кружница — Википедија

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color:#444; font-size:1.15em; line-height:1.5;\"\u003E\u003Cdiv style=\"padding-left:8px; padding-right:8px;\"\u003EПридружите се \u003Cb\u003E\u003Ca href=\"/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%98%D0%B0:%D0%A3%D1%80%D0%B5%D1%92%D0%B8%D0%B2%D0%B0%D1%87%D0%BA%D0%B8_%D0%BC%D0%B0%D1%80%D0%B0%D1%82%D0%BE%D0%BD_%D0%92%D0%B8%D0%BA%D0%B8_%D0%B2%D0%BE%D0%BB%D0%B8_%D1%98%D0%B0%D0%B2%D0%BD%D1%83_%D1%83%D0%BC%D0%B5%D1%82%D0%BD%D0%BE%D1%81%D1%82_%D0%B8_%D0%B3%D1%80%D0%BE%D0%B1%D0%BD%D0%B0_%D0%BE%D0%B1%D0%B5%D0%BB%D0%B5%D0%B6%D1%98%D0%B0_2024.\" title=\"Википедија:Уређивачки маратон Вики воли јавну уметност и гробна обележја 2024.\"\u003EУређивачком маратону Вики воли јавну уметност и гробна обележја 2024\u003C/a\u003E\u003C/b\u003E.\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></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="Сајт"> <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="Садржај" 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">Садржај</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">помери на страну</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">сакриј</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">Почетак</div> </a> </li> <li id="toc-Дефиниције" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Дефиниције"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Дефиниције</span> </div> </a> <button aria-controls="toc-Дефиниције-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Садржај одељка Дефиниције</span> </button> <ul id="toc-Дефиниције-sublist" class="vector-toc-list"> <li id="toc-Остале_дефиниције" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Остале_дефиниције"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Остале дефиниције</span> </div> </a> <ul id="toc-Остале_дефиниције-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Елементарна_(Еуклидска)_геометрија" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Елементарна_(Еуклидска)_геометрија"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Елементарна (Еуклидска) геометрија</span> </div> </a> <ul id="toc-Елементарна_(Еуклидска)_геометрија-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Растојање_тачке_од_кружнице" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Растојање_тачке_од_кружнице"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Растојање тачке од кружнице</span> </div> </a> <ul id="toc-Растојање_тачке_од_кружнице-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Заједничке_тачке_кружница" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Заједничке_тачке_кружница"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Заједничке тачке кружница</span> </div> </a> <button aria-controls="toc-Заједничке_тачке_кружница-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Садржај одељка Заједничке тачке кружница</span> </button> <ul id="toc-Заједничке_тачке_кружница-sublist" class="vector-toc-list"> <li id="toc-Пресек_кружница_празан_скуп" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Пресек_кружница_празан_скуп"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Пресек кружница празан скуп</span> </div> </a> <ul id="toc-Пресек_кружница_празан_скуп-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Тангента_кружнице" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Тангента_кружнице"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Тангента кружнице</span> </div> </a> <button aria-controls="toc-Тангента_кружнице-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Садржај одељка Тангента кружнице</span> </button> <ul id="toc-Тангента_кружнице-sublist" class="vector-toc-list"> <li id="toc-Тангирање_кружница" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Тангирање_кружница"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Тангирање кружница</span> </div> </a> <ul id="toc-Тангирање_кружница-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Пресек_кружница" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Пресек_кружница"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Пресек кружница</span> </div> </a> <ul id="toc-Пресек_кружница-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Полара_кружнице" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Полара_кружнице"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Полара кружнице</span> </div> </a> <button aria-controls="toc-Полара_кружнице-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Садржај одељка Полара кружнице</span> </button> <ul id="toc-Полара_кружнице-sublist" class="vector-toc-list"> <li id="toc-Аполонијева_кружница" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Аполонијева_кружница"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Аполонијева кружница</span> </div> </a> <ul id="toc-Аполонијева_кружница-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Кружнице_у_-{p}--нормама_и_бројеви_'"`UNIQ--postMath-0000004D-QINU`"'" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Кружнице_у_-{p}--нормама_и_бројеви_'"`UNIQ--postMath-0000004D-QINU`"'"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Кружнице у p-нормама и бројеви '"`UNIQ--postMath-0000004D-QINU`"'</span> </div> </a> <ul id="toc-Кружнице_у_-{p}--нормама_и_бројеви_'"`UNIQ--postMath-0000004D-QINU`"'-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Референце" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Референце"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Референце</span> </div> </a> <ul id="toc-Референце-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Литература" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Литература"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Литература</span> </div> </a> <ul id="toc-Литература-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Спољашње_везе" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Спољашње_везе"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Спољашње везе</span> </div> </a> <ul id="toc-Спољашње_везе-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="Садржај" 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="Прикажи/сакриј садржај" > <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">Прикажи/сакриј садржај</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">Кружница</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="Чланак на другим језицима. Доступан на: 145" > <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-145" 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">145 језика</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Sirkel" title="Sirkel — африканс" lang="af" hreflang="af" data-title="Sirkel" data-language-autonym="Afrikaans" data-language-local-name="африканс" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Kreis_(Geometrie)" title="Kreis (Geometrie) — немачки (Швајцарска)" lang="gsw" hreflang="gsw" data-title="Kreis (Geometrie)" data-language-autonym="Alemannisch" data-language-local-name="немачки (Швајцарска)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%AD%E1%89%A5" title="ክብ — амхарски" lang="am" hreflang="am" data-title="ክብ" data-language-autonym="አማርኛ" data-language-local-name="амхарски" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar badge-Q17437796 badge-featuredarticle mw-list-item" title="сјајан чланак"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9" title="دائرة — арапски" lang="ar" hreflang="ar" data-title="دائرة" data-language-autonym="العربية" data-language-local-name="арапски" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Circumferencia" title="Circumferencia — арагонски" lang="an" hreflang="an" data-title="Circumferencia" data-language-autonym="Aragonés" data-language-local-name="арагонски" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-roa-rup mw-list-item"><a href="https://roa-rup.wikipedia.org/wiki/%C8%9Aerc%C4%BEiu" title="Țercľiu — цинцарски" lang="rup" hreflang="rup" data-title="Țercľiu" data-language-autonym="Armãneashti" data-language-local-name="цинцарски" class="interlanguage-link-target"><span>Armãneashti</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%AF%D8%A7%D9%8A%D8%B1%D9%87" title="دايره — Egyptian Arabic" lang="arz" hreflang="arz" data-title="دايره" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="বৃত্ত — асамски" lang="as" hreflang="as" data-title="বৃত্ত" data-language-autonym="অসমীয়া" data-language-local-name="асамски" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Circunferencia" title="Circunferencia — астуријски" lang="ast" hreflang="ast" data-title="Circunferencia" data-language-autonym="Asturianu" data-language-local-name="астуријски" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Muyu" title="Muyu — ајмара" lang="ay" hreflang="ay" data-title="Muyu" data-language-autonym="Aymar aru" data-language-local-name="ајмара" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87evr%C9%99" title="Çevrə — азербејџански" lang="az" hreflang="az" data-title="Çevrə" data-language-autonym="Azərbaycanca" data-language-local-name="азербејџански" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AF%D8%A7%DB%8C%D8%B1%D9%87" title="دایره — South Azerbaijani" lang="azb" hreflang="azb" data-title="دایره" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lingkaran" title="Lingkaran — индонежански" lang="id" hreflang="id" data-title="Lingkaran" data-language-autonym="Bahasa Indonesia" data-language-local-name="индонежански" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Bulatan" title="Bulatan — малајски" lang="ms" hreflang="ms" data-title="Bulatan" data-language-autonym="Bahasa Melayu" data-language-local-name="малајски" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%BA%D1%80%D1%8A%D0%B6%D0%BD%D0%BE%D1%81%D1%82" title="Окръжност — бугарски" lang="bg" hreflang="bg" data-title="Окръжност" data-language-autonym="Български" data-language-local-name="бугарски" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/%C3%8E%E2%81%BF-h%C3%AAng" title="Îⁿ-hêng — Minnan" lang="nan" hreflang="nan" data-title="Îⁿ-hêng" 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><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D3%98%D0%B9%D0%BB%D3%99%D0%BD%D3%99" title="Әйләнә — башкирски" lang="ba" hreflang="ba" data-title="Әйләнә" data-language-autonym="Башҡортса" data-language-local-name="башкирски" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Акружнасць — белоруски" lang="be" hreflang="be" data-title="Акружнасць" data-language-autonym="Беларуская" data-language-local-name="белоруски" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%BA%D1%80%D1%83%D0%B6%D1%8B%D0%BD%D0%B0" title="Акружына — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="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-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Bilog" title="Bilog — Central Bikol" lang="bcl" hreflang="bcl" data-title="Bilog" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="বৃত্ত — бенгалски" lang="bn" hreflang="bn" data-title="বৃত্ত" data-language-autonym="বাংলা" data-language-local-name="бенгалски" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A6%E0%BE%92%E0%BD%BC%E0%BD%A2%E0%BC%8B%E0%BD%90%E0%BD%B2%E0%BD%82%E0%BC%8B" title="སྒོར་ཐིག་ — тибетански" lang="bo" hreflang="bo" data-title="སྒོར་ཐིག་" data-language-autonym="བོད་ཡིག" data-language-local-name="тибетански" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica — босански" lang="bs" hreflang="bs" data-title="Kružnica" data-language-autonym="Bosanski" data-language-local-name="босански" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Kelc%27h" title="Kelc'h — бретонски" lang="br" hreflang="br" data-title="Kelc'h" data-language-autonym="Brezhoneg" data-language-local-name="бретонски" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Circumfer%C3%A8ncia" title="Circumferència — каталонски" lang="ca" hreflang="ca" data-title="Circumferència" data-language-autonym="Català" data-language-local-name="каталонски" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Denderedzwa" title="Denderedzwa — шона" lang="sn" hreflang="sn" data-title="Denderedzwa" data-language-autonym="ChiShona" data-language-local-name="шона" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cylch" title="Cylch — велшки" lang="cy" hreflang="cy" data-title="Cylch" data-language-autonym="Cymraeg" data-language-local-name="велшки" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C3%87%D0%B0%D0%B2%D1%80%D0%B0%D0%BA%C4%83%D1%88" title="Çавракăш — чувашки" lang="cv" hreflang="cv" data-title="Çавракăш" data-language-autonym="Чӑвашла" data-language-local-name="чувашки" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kru%C5%BEnice" title="Kružnice — чешки" lang="cs" hreflang="cs" data-title="Kružnice" data-language-autonym="Čeština" data-language-local-name="чешки" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Cirkel" title="Cirkel — дански" lang="da" hreflang="da" data-title="Cirkel" data-language-autonym="Dansk" data-language-local-name="дански" 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/Kreis" title="Kreis — немачки" lang="de" hreflang="de" data-title="Kreis" data-language-autonym="Deutsch" data-language-local-name="немачки" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Cera_krejza" title="Cera krejza — доњолужичкосрпски" lang="dsb" hreflang="dsb" data-title="Cera krejza" data-language-autonym="Dolnoserbski" data-language-local-name="доњолужичкосрпски" class="interlanguage-link-target"><span>Dolnoserbski</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ringjoon" title="Ringjoon — естонски" lang="et" hreflang="et" data-title="Ringjoon" data-language-autonym="Eesti" data-language-local-name="естонски" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%8D%CE%BA%CE%BB%CE%BF%CF%82" title="Κύκλος — грчки" lang="el" hreflang="el" data-title="Κύκλος" data-language-autonym="Ελληνικά" data-language-local-name="грчки" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Ser%C4%87_(giometr%C3%ACa)" title="Serć (giometrìa) — Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Serć (giometrìa)" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Circle" title="Circle — енглески" lang="en" hreflang="en" data-title="Circle" data-language-autonym="English" data-language-local-name="енглески" 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/C%C3%ADrculo" title="Círculo — шпански" lang="es" hreflang="es" data-title="Círculo" data-language-autonym="Español" data-language-local-name="шпански" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Cirklo" title="Cirklo — есперанто" lang="eo" hreflang="eo" data-title="Cirklo" data-language-autonym="Esperanto" data-language-local-name="есперанто" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zirkulu" title="Zirkulu — баскијски" lang="eu" hreflang="eu" data-title="Zirkulu" data-language-autonym="Euskara" data-language-local-name="баскијски" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa badge-Q17437798 badge-goodarticle mw-list-item" title="добар чланак"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%A7%DB%8C%D8%B1%D9%87" title="دایره — персијски" lang="fa" hreflang="fa" data-title="دایره" data-language-autonym="فارسی" data-language-local-name="персијски" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Circle" title="Circle — Fiji Hindi" lang="hif" hreflang="hif" data-title="Circle" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Sirkul" title="Sirkul — фарски" lang="fo" hreflang="fo" data-title="Sirkul" data-language-autonym="Føroyskt" data-language-local-name="фарски" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cercle" title="Cercle — француски" lang="fr" hreflang="fr" data-title="Cercle" data-language-autonym="Français" data-language-local-name="француски" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Ciorcal" title="Ciorcal — ирски" lang="ga" hreflang="ga" data-title="Ciorcal" data-language-autonym="Gaeilge" data-language-local-name="ирски" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Kiarkyl" title="Kiarkyl — манкс" lang="gv" hreflang="gv" data-title="Kiarkyl" data-language-autonym="Gaelg" data-language-local-name="манкс" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Cearcall" title="Cearcall — шкотски гелски" lang="gd" hreflang="gd" data-title="Cearcall" data-language-autonym="Gàidhlig" data-language-local-name="шкотски гелски" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Circunferencia" title="Circunferencia — галицијски" lang="gl" hreflang="gl" data-title="Circunferencia" data-language-autonym="Galego" data-language-local-name="галицијски" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%9C%93%E5%BD%A2" title="圓形 — Gan" lang="gan" hreflang="gan" data-title="圓形" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B5%E0%AA%B0%E0%AB%8D%E0%AA%A4%E0%AB%81%E0%AA%B3" title="વર્તુળ — гуџарати" lang="gu" hreflang="gu" data-title="વર્તુળ" data-language-autonym="ગુજરાતી" data-language-local-name="гуџарати" class="interlanguage-link-target"><span>ગુજરાતી</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" title="מעגל — хебрејски" lang="he" hreflang="he" data-title="מעגל" data-language-autonym="עברית" data-language-local-name="хебрејски" 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%B5%E0%A5%83%E0%A4%A4%E0%A5%8D%E0%A4%A4" title="वृत्त — хинди" lang="hi" hreflang="hi" data-title="वृत्त" data-language-autonym="हिन्दी" data-language-local-name="хинди" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica — горњолужичкосрпски" lang="hsb" hreflang="hsb" data-title="Kružnica" data-language-autonym="Hornjoserbsce" data-language-local-name="горњолужичкосрпски" class="interlanguage-link-target"><span>Hornjoserbsce</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica — хрватски" lang="hr" hreflang="hr" data-title="Kružnica" data-language-autonym="Hrvatski" data-language-local-name="хрватски" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%87%D6%80%D5%BB%D5%A1%D5%B6%D5%A1%D5%A3%D5%AB%D5%AE" title="Շրջանագիծ — јерменски" lang="hy" hreflang="hy" data-title="Շրջանագիծ" data-language-autonym="Հայերեն" data-language-local-name="јерменски" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Cirklo" title="Cirklo — идо" lang="io" hreflang="io" data-title="Cirklo" data-language-autonym="Ido" data-language-local-name="идо" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Circulo" title="Circulo — интерлингва" lang="ia" hreflang="ia" data-title="Circulo" data-language-autonym="Interlingua" data-language-local-name="интерлингва" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Isazinge" title="Isazinge — коса" lang="xh" hreflang="xh" data-title="Isazinge" data-language-autonym="IsiXhosa" data-language-local-name="коса" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hringur_(r%C3%BAmfr%C3%A6%C3%B0i)" title="Hringur (rúmfræði) — исландски" lang="is" hreflang="is" data-title="Hringur (rúmfræði)" data-language-autonym="Íslenska" data-language-local-name="исландски" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Circonferenza" title="Circonferenza — италијански" lang="it" hreflang="it" data-title="Circonferenza" data-language-autonym="Italiano" data-language-local-name="италијански" 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%86_(%E6%95%B0%E5%AD%A6)" title="円 (数学) — јапански" lang="ja" hreflang="ja" data-title="円 (数学)" data-language-autonym="日本語" data-language-local-name="јапански" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Bunderan" title="Bunderan — јавански" lang="jv" hreflang="jv" data-title="Bunderan" data-language-autonym="Jawa" data-language-local-name="јавански" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AC%E1%83%A0%E1%83%94%E1%83%AC%E1%83%98%E1%83%A0%E1%83%98" title="წრეწირი — грузијски" lang="ka" hreflang="ka" data-title="წრეწირი" data-language-autonym="ქართული" data-language-local-name="грузијски" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A8%D0%B5%D2%A3%D0%B1%D0%B5%D1%80" title="Шеңбер — казашки" lang="kk" hreflang="kk" data-title="Шеңбер" data-language-autonym="Қазақша" data-language-local-name="казашки" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Kylgh" title="Kylgh — корнволски" lang="kw" hreflang="kw" data-title="Kylgh" data-language-autonym="Kernowek" data-language-local-name="корнволски" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B9%D0%BB%D0%B0%D0%BD%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Айлана (математика) — киргиски" lang="ky" hreflang="ky" data-title="Айлана (математика)" data-language-autonym="Кыргызча" data-language-local-name="киргиски" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Duara" title="Duara — свахили" lang="sw" hreflang="sw" data-title="Duara" data-language-autonym="Kiswahili" data-language-local-name="свахили" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9A%E1%9E%84%E1%9F%92%E1%9E%9C%E1%9E%84%E1%9F%8B" title="រង្វង់ — кмерски" lang="km" hreflang="km" data-title="រង្វង់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="кмерски" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B3%83%E0%B2%A4%E0%B3%8D%E0%B2%A4" title="ವೃತ್ತ — канада" lang="kn" hreflang="kn" data-title="ವೃತ್ತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="канада" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90_(%EA%B8%B0%ED%95%98%ED%95%99)" title="원 (기하학) — корејски" lang="ko" hreflang="ko" data-title="원 (기하학)" data-language-autonym="한국어" data-language-local-name="корејски" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/S%C3%A8k_(non)" title="Sèk (non) — хаићански" lang="ht" hreflang="ht" data-title="Sèk (non)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="хаићански" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Serk" title="Serk — Guianan Creole" lang="gcr" hreflang="gcr" data-title="Serk" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Gilover" title="Gilover — курдски" lang="ku" hreflang="ku" data-title="Gilover" data-language-autonym="Kurdî" data-language-local-name="курдски" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Circulus" title="Circulus — латински" lang="la" hreflang="la" data-title="Circulus" data-language-autonym="Latina" data-language-local-name="латински" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Ri%C5%86%C4%B7a_l%C4%ABnija" title="Riņķa līnija — летонски" lang="lv" hreflang="lv" data-title="Riņķa līnija" data-language-autonym="Latviešu" data-language-local-name="летонски" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Krees_(Geometrie)" title="Krees (Geometrie) — луксембуршки" lang="lb" hreflang="lb" data-title="Krees (Geometrie)" data-language-autonym="Lëtzebuergesch" data-language-local-name="луксембуршки" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Apskritimas" title="Apskritimas — литвански" lang="lt" hreflang="lt" data-title="Apskritimas" data-language-autonym="Lietuvių" data-language-local-name="литвански" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Cirkel" title="Cirkel — лимбуршки" lang="li" hreflang="li" data-title="Cirkel" data-language-autonym="Limburgs" data-language-local-name="лимбуршки" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Sirculo" title="Sirculo — Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Sirculo" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/S%C3%A9rcc" title="Sércc — ломбард" lang="lmo" hreflang="lmo" data-title="Sércc" data-language-autonym="Lombard" data-language-local-name="ломбард" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%B6r_(geometria)" title="Kör (geometria) — мађарски" lang="hu" hreflang="hu" data-title="Kör (geometria)" data-language-autonym="Magyar" data-language-local-name="мађарски" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="сјајан чланак"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0" title="Кружница — македонски" lang="mk" hreflang="mk" data-title="Кружница" data-language-autonym="Македонски" data-language-local-name="македонски" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Faribolana" title="Faribolana — малгашки" lang="mg" hreflang="mg" data-title="Faribolana" data-language-autonym="Malagasy" data-language-local-name="малгашки" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Lingkaran" title="Lingkaran — минангкабау" lang="min" hreflang="min" data-title="Lingkaran" data-language-autonym="Minangkabau" data-language-local-name="минангкабау" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B5%83%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%82" title="വൃത്തം — малајалам" lang="ml" hreflang="ml" data-title="വൃത്തം" data-language-autonym="മലയാളം" data-language-local-name="малајалам" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B9%D1%80%D0%BE%D0%B3" title="Тойрог — монголски" lang="mn" hreflang="mn" data-title="Тойрог" data-language-autonym="Монгол" data-language-local-name="монголски" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A5%81%E0%A4%B3" title="वर्तुळ — марати" lang="mr" hreflang="mr" data-title="वर्तुळ" data-language-autonym="मराठी" data-language-local-name="марати" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%85%E1%80%80%E1%80%BA%E1%80%9D%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8" title="စက်ဝိုင်း — бурмански" lang="my" hreflang="my" data-title="စက်ဝိုင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="бурмански" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Iwirini" title="Iwirini — фиџијски" lang="fj" hreflang="fj" data-title="Iwirini" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="фиџијски" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A5%83%E0%A4%A4" title="वृत — непалски" lang="ne" hreflang="ne" data-title="वृत" data-language-autonym="नेपाली" data-language-local-name="непалски" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Cirkel" title="Cirkel — холандски" lang="nl" hreflang="nl" data-title="Cirkel" data-language-autonym="Nederlands" data-language-local-name="холандски" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%9A%E0%A4%BE%E0%A4%95%E0%A4%83" title="चाकः — невари" lang="new" hreflang="new" data-title="चाकः" data-language-autonym="नेपाल भाषा" data-language-local-name="невари" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kreis_(Geometrii)" title="Kreis (Geometrii) — севернофризијски" lang="frr" hreflang="frr" data-title="Kreis (Geometrii)" data-language-autonym="Nordfriisk" data-language-local-name="севернофризијски" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-pih mw-list-item"><a href="https://pih.wikipedia.org/wiki/Sirkil" title="Sirkil — Norfuk / Pitkern" lang="pih" hreflang="pih" data-title="Sirkil" data-language-autonym="Norfuk / Pitkern" data-language-local-name="Norfuk / Pitkern" class="interlanguage-link-target"><span>Norfuk / Pitkern</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sirkel" title="Sirkel — норвешки букмол" lang="nb" hreflang="nb" data-title="Sirkel" data-language-autonym="Norsk bokmål" data-language-local-name="норвешки букмол" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Sirkel" title="Sirkel — норвешки нинорск" lang="nn" hreflang="nn" data-title="Sirkel" data-language-autonym="Norsk nynorsk" data-language-local-name="норвешки нинорск" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Cercle" title="Cercle — окситански" lang="oc" hreflang="oc" data-title="Cercle" data-language-autonym="Occitan" data-language-local-name="окситански" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9E%D2%A5%D0%B3%D0%BE" title="Оҥго — Eastern Mari" lang="mhr" hreflang="mhr" data-title="Оҥго" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AC%E0%AD%83%E0%AC%A4%E0%AD%8D%E0%AC%A4" title="ବୃତ୍ତ — одија" lang="or" hreflang="or" data-title="ବୃତ୍ତ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="одија" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Geengoo" title="Geengoo — оромо" lang="om" hreflang="om" data-title="Geengoo" data-language-autonym="Oromoo" data-language-local-name="оромо" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Aylana" title="Aylana — узбечки" lang="uz" hreflang="uz" data-title="Aylana" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="узбечки" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9A%E0%A9%B1%E0%A8%95%E0%A8%B0" title="ਚੱਕਰ — пенџапски" lang="pa" hreflang="pa" data-title="ਚੱਕਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="пенџапски" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Krink" title="Krink — нисконемачки" lang="nds" hreflang="nds" data-title="Krink" data-language-autonym="Plattdüütsch" data-language-local-name="нисконемачки" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%DB%81" title="دائرہ — Western Punjabi" lang="pnb" hreflang="pnb" data-title="دائرہ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Okr%C4%85g" title="Okrąg — пољски" lang="pl" hreflang="pl" data-title="Okrąg" data-language-autonym="Polski" data-language-local-name="пољски" 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/Circunfer%C3%AAncia" title="Circunferência — португалски" lang="pt" hreflang="pt" data-title="Circunferência" data-language-autonym="Português" data-language-local-name="португалски" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DA%AF%D8%B1%D8%AF%DA%A9%D9%87" title="گردکه — паштунски" lang="ps" hreflang="ps" data-title="گردکه" data-language-autonym="پښتو" data-language-local-name="паштунски" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Soerkl" title="Soerkl — Jamaican Creole English" lang="jam" hreflang="jam" data-title="Soerkl" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Cerc" title="Cerc — румунски" lang="ro" hreflang="ro" data-title="Cerc" data-language-autonym="Română" data-language-local-name="румунски" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/P%27allta_muyu" title="P'allta muyu — кечуа" lang="qu" hreflang="qu" data-title="P'allta muyu" data-language-autonym="Runa Simi" data-language-local-name="кечуа" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B3" title="Круг — Rusyn" lang="rue" hreflang="rue" data-title="Круг" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Окружность — руски" lang="ru" hreflang="ru" data-title="Окружность" data-language-autonym="Русский" data-language-local-name="руски" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Raing" title="Raing — шкотски" lang="sco" hreflang="sco" data-title="Raing" data-language-autonym="Scots" data-language-local-name="шкотски" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AF%D9%88%D9%84" title="گول — синди" lang="sd" hreflang="sd" data-title="گول" data-language-autonym="سنڌي" data-language-local-name="синди" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rrethi" title="Rrethi — албански" lang="sq" hreflang="sq" data-title="Rrethi" data-language-autonym="Shqip" data-language-local-name="албански" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/C%C3%ACrculu_(giometr%C3%ACa)" title="Cìrculu (giometrìa) — сицилијански" lang="scn" hreflang="scn" data-title="Cìrculu (giometrìa)" data-language-autonym="Sicilianu" data-language-local-name="сицилијански" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Circle" title="Circle — Simple English" lang="en-simple" hreflang="en-simple" data-title="Circle" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica — словачки" lang="sk" hreflang="sk" data-title="Kružnica" data-language-autonym="Slovenčina" data-language-local-name="словачки" 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/Kro%C5%BEnica" title="Krožnica — словеначки" lang="sl" hreflang="sl" data-title="Krožnica" data-language-autonym="Slovenščina" data-language-local-name="словеначки" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Goobo" title="Goobo — сомалски" lang="so" hreflang="so" data-title="Goobo" data-language-autonym="Soomaaliga" data-language-local-name="сомалски" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%D8%A7%D8%B2%D9%86%DB%95_(%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95)" title="بازنە (ئەندازە) — централни курдски" lang="ckb" hreflang="ckb" data-title="بازنە (ئەندازە)" data-language-autonym="کوردی" data-language-local-name="централни курдски" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kru%C5%BEnica" title="Kružnica — српскохрватски" lang="sh" hreflang="sh" data-title="Kružnica" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="српскохрватски" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Bunderan_(%C3%A9lmu_ukur)" title="Bunderan (élmu ukur) — сундански" lang="su" hreflang="su" data-title="Bunderan (élmu ukur)" data-language-autonym="Sunda" data-language-local-name="сундански" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ympyr%C3%A4" title="Ympyrä — фински" lang="fi" hreflang="fi" data-title="Ympyrä" data-language-autonym="Suomi" data-language-local-name="фински" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Cirkel" title="Cirkel — шведски" lang="sv" hreflang="sv" data-title="Cirkel" data-language-autonym="Svenska" data-language-local-name="шведски" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="வட்டம் — тамилски" lang="ta" hreflang="ta" data-title="வட்டம்" data-language-autonym="தமிழ்" data-language-local-name="тамилски" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Bilog" title="Bilog — тагалог" lang="tl" hreflang="tl" data-title="Bilog" data-language-autonym="Tagalog" data-language-local-name="тагалог" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tawinest" title="Tawinest — кабиле" lang="kab" hreflang="kab" data-title="Tawinest" data-language-autonym="Taqbaylit" data-language-local-name="кабиле" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D3%98%D0%B9%D0%BB%D3%99%D0%BD%D3%99" title="Әйләнә — татарски" lang="tt" hreflang="tt" data-title="Әйләнә" data-language-autonym="Татарча / tatarça" data-language-local-name="татарски" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B1%83%E0%B0%A4%E0%B1%8D%E0%B0%A4%E0%B0%AE%E0%B1%81" title="వృత్తము — телугу" lang="te" hreflang="te" data-title="వృత్తము" data-language-autonym="తెలుగు" data-language-local-name="телугу" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%A7%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%A1" title="รูปวงกลม — тајски" lang="th" hreflang="th" data-title="รูปวงกลม" data-language-autonym="ไทย" data-language-local-name="тајски" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%B0%E1%BB%9Dng_tr%C3%B2n" title="Đường tròn — вијетнамски" lang="vi" hreflang="vi" data-title="Đường tròn" data-language-autonym="Tiếng Việt" data-language-local-name="вијетнамски" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87ember" title="Çember — турски" lang="tr" hreflang="tr" data-title="Çember" data-language-autonym="Türkçe" data-language-local-name="турски" 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%9A%D0%BE%D0%BB%D0%BE" title="Коло — украјински" lang="uk" hreflang="uk" data-title="Коло" data-language-autonym="Українська" data-language-local-name="украјински" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%DB%81" title="دائرہ — урду" lang="ur" hreflang="ur" data-title="دائرہ" data-language-autonym="اردو" data-language-local-name="урду" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-za mw-list-item"><a href="https://za.wikipedia.org/wiki/Luenz" title="Luenz — џуаншки" lang="za" hreflang="za" data-title="Luenz" data-language-autonym="Vahcuengh" data-language-local-name="џуаншки" class="interlanguage-link-target"><span>Vahcuengh</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Sercio" title="Sercio — Venetian" lang="vec" hreflang="vec" data-title="Sercio" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Ts%C3%B5%C3%B5rjuun" title="Tsõõrjuun — Võro" lang="vro" hreflang="vro" data-title="Tsõõrjuun" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Lidong" title="Lidong — варајски" lang="war" hreflang="war" data-title="Lidong" data-language-autonym="Winaray" data-language-local-name="варајски" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9C%86" title="圆 — ву кинески" lang="wuu" hreflang="wuu" data-title="圆" data-language-autonym="吴语" data-language-local-name="ву кинески" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%A8%D7%99%D7%99%D7%96" title="קרייז — јидиш" lang="yi" hreflang="yi" data-title="קרייז" data-language-autonym="ייִדיש" data-language-local-name="јидиш" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%92b%C3%ACr%C3%ADpo" title="Òbìrípo — јоруба" lang="yo" hreflang="yo" data-title="Òbìrípo" data-language-autonym="Yorùbá" data-language-local-name="јоруба" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9C%86" title="圆 — кинески" lang="zh" hreflang="zh" data-title="圆" data-language-autonym="中文" data-language-local-name="кинески" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%9C%93" title="圓 — Literary Chinese" lang="lzh" hreflang="lzh" data-title="圓" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9C%93%E5%BD%A2" title="圓形 — кантонски" lang="yue" hreflang="yue" data-title="圓形" data-language-autonym="粵語" data-language-local-name="кантонски" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Apskr%C4%97t%C4%97ms" title="Apskrėtėms — Samogitian" lang="sgs" hreflang="sgs" data-title="Apskrėtėms" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</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/Q17278#sitelinks-wikipedia" title="Уреди међујезичке везе" class="wbc-editpage">Уреди везе</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="Именски простори"> <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" 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class="mw-list-item"><a href="/w/index.php?title=%D0%9F%D0%BE%D1%81%D0%B5%D0%B1%D0%BD%D0%BE:DownloadAsPdf&page=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=show-download-screen"><span>Преузми у PDF-у</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="Одштампајте ову страницу [p]" accesskey="p"><span>Одштампај</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> На другим пројектима </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Circle_area" hreflang="en"><span>Викиостава</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q17278" title="Веза ка ставци на спремишту података [g]" accesskey="g"><span>Ставка на Википодацима</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Алатке странице"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Изглед"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Изглед</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">помери на страну</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">сакриј</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">С Википедије, слободне енциклопедије</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="sr" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r28440203">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3;">Круг</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/%D0%94%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Circle-withsegments.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Circle-withsegments.svg/220px-Circle-withsegments.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Circle-withsegments.svg/330px-Circle-withsegments.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Circle-withsegments.svg/440px-Circle-withsegments.svg.png 2x" data-file-width="726" data-file-height="726" /></a></span><div class="infobox-caption">Круг (црно), који се одређује његовим обимом (<i>C</i>), пречником (<i>D</i>) у цијану, и полупречником (<i>R</i>) у црвеном; његов центар (<i>O</i>) је у маргенти.</div></td></tr></tbody></table> <p><b>Кружница</b> је <a href="/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика">математички</a> <a href="/wiki/%D0%9F%D0%BE%D1%98%D0%B0%D0%BC" title="Појам">појам</a> који се користи у <a href="/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Геометрија">геометрији</a> и није <a href="/wiki/%D0%A1%D0%B8%D0%BD%D0%BE%D0%BD%D0%B8%D0%BC" title="Синоним">синоним</a> за <a href="/wiki/%D0%9A%D1%80%D1%83%D0%B3" title="Круг">круг</a>. Уобичајено је да се кружница зове <a href="/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%B0" title="Крива">линија</a> коју описује <a href="/wiki/%C5%A0estar" title="Šestar">шестар</a>, а круг је <a href="/wiki/%D0%9F%D0%BE%D0%B2%D1%80%D1%88%D0%B8%D0%BD%D0%B0" title="Површина">површина</a> унутар кружнице. Тако кружница има своју <a href="/wiki/%D0%94%D1%83%D0%B6%D0%B8%D0%BD%D0%B0" title="Дужина">дужину</a>, која се често зове <a href="/wiki/%D0%9E%D0%B1%D0%B8%D0%BC" title="Обим">обим</a>, а круг има површину. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Дефиниције"><span id=".D0.94.D0.B5.D1.84.D0.B8.D0.BD.D0.B8.D1.86.D0.B8.D1.98.D0.B5"></span>Дефиниције</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=1" title="Уредите одељак „Дефиниције”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=1" title="Уреди извор одељка: Дефиниције"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Кружница</b> је затворена <a href="/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%B0" title="Крива">крива линија</a> у <a href="/wiki/%D0%A0%D0%B0%D0%B2%D0%B0%D0%BD" title="Раван">равни</a> чије све тачке леже на истом одстојању од неке тачке <i><b>О</b></i> у истој овој равни и која се зове центар кружнице. Одстојање сваке тачке кружнице од њеног центра мери се сегментом <a href="/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%B0_(%D0%BB%D0%B8%D0%BD%D0%B8%D1%98%D0%B0)" title="Права (линија)">праве</a> који се назива <a href="/wiki/%D0%9F%D0%BE%D0%BB%D1%83%D0%BF%D1%80%D0%B5%D1%87%D0%BD%D0%B8%D0%BA" title="Полупречник">полупречник</a> (радијус) <i><b>r</b></i>. Кружница <i><b>k</b></i> с центром <i><b>О</b></i> и полупречником <i><b>r</b></i> означава се <i><b>k(O,r)</b></i>, понекад са <i><b>O(r)</b></i>. </p><p><b>Кружница</b> са центром <i><b>О</b></i> и полупречником <i><b>r</b></i> може се дефинисати као <a href="/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D1%81%D0%BA%D0%BE_%D0%BC%D0%B5%D1%81%D1%82%D0%BE_%D1%82%D0%B0%D1%87%D0%B0%D0%BA%D0%B0" title="Геометријско место тачака">геометријско место тачака</a> у <a href="/wiki/%D0%A0%D0%B0%D0%B2%D0%B0%D0%BD" title="Раван">равни</a> на датом одстојању <i><b>r</b></i> од дате тачке <i><b>О</b></i> која лежи у истој равни. </p><p><b>Једначина кружнице</b> у правоуглим Декартовим координатама гласи: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-p)^{2}+(y-q)^{2}=r^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>q</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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-p)^{2}+(y-q)^{2}=r^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7dfbb201dfdeef2f341609907b1f01799ea8dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.561ex; height:3.176ex;" alt="{\displaystyle (x-p)^{2}+(y-q)^{2}=r^{2}\,}"></span>, </p><p>где су <i><b>(p,q)</b></i> координате центра, а <i><b>r</b></i> полупречник. Из претходне једначине следи да је кружница <a href="/w/index.php?title=%D0%9A%D1%80%D0%B8%D0%B2%D0%B0_%D0%B4%D1%80%D1%83%D0%B3%D0%BE%D0%B3_%D1%80%D0%B5%D0%B4%D0%B0&action=edit&redlink=1" class="new" title="Крива другог реда (страница не постоји)">крива другог реда</a>. Претходна <a href="/wiki/%D0%88%D0%B5%D0%B4%D0%BD%D0%B0%D1%87%D0%B8%D0%BD%D0%B0" title="Једначина">једначина</a> кружнице се користи у решавању конструктивних задатака, у графичком решавању једначина и неједнакости. Ова је једначина другог <a href="/wiki/%D0%A0%D0%B5%D0%B4_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ред (математика)">реда</a>. Једначина кружнице може се написати и на следећи начин </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{2}}{r^{2}}}+{\frac {y^{2}}{r^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{r^{2}}}+{\frac {y^{2}}{r^{2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5c477c2f52a31eae92555b545b26aa02249f46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.372ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{r^{2}}}+{\frac {y^{2}}{r^{2}}}=1}"></span>. Ово је сегментна једначина.</dd></dl></dd></dl> <p>Кружница са средиштем у тачки <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(p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9f0a69ad929e6f887e8a18ddabc0f15b90987e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.581ex; height:2.843ex;" alt="{\displaystyle S(p,q)}"></span> и полупречником <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> одређена је једначином: </p> <dl><dd><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 (x-p)^{2}+(y-q)^{2}={r^{2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-p)^{2}+(y-q)^{2}={r^{2}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724d5fc63303178a0eff32a2d43b78daf6bbc473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.561ex; height:3.176ex;" alt="{\displaystyle (x-p)^{2}+(y-q)^{2}={r^{2}}\,}"></span></dd></dl></dd></dl> <p>или </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(x-p)^{2}}{r^{2}}}+{\frac {(y-q)^{2}}{r^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(x-p)^{2}}{r^{2}}}+{\frac {(y-q)^{2}}{r^{2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0015c5c86feeb8711095dff66a74811c8223bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.906ex; height:6.176ex;" alt="{\displaystyle {\frac {(x-p)^{2}}{r^{2}}}+{\frac {(y-q)^{2}}{r^{2}}}=1}"></span></dd></dl></dd></dl> <p>У свакој тачки кружнице њена <a href="/wiki/Zakrivljenost" title="Zakrivljenost">кривина</a> је константна, једнака <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 {1}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd462218ef3cc25ed3b835b52af9b951d54edb13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{r}}}"></span>. <a href="/wiki/%D0%A2%D0%B0%D0%BD%D0%B3%D0%B5%D0%BD%D1%82%D0%B0" title="Тангента">Тангента</a> на кружницу је нормална на полупречник у тачки додира. </p><p>Обим кружнице <i><b>O(r)</b></i> је <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 2r\pi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>r</mi> <mi>π</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2r\pi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c05b63a059b0b229f657a491f27a9698c6e28d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.93ex; height:2.176ex;" alt="{\displaystyle 2r\pi \,}"></span>, а кружница се назива и <b>периферијом <a href="/wiki/%D0%9A%D1%80%D1%83%D0%B3" title="Круг">круга</a></b>. </p><p>Површина омеђена кружницом је <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}\pi \,}"> <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> <mi>π</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}\pi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd67c0d8f4d9025d589a18c46eda9643ca4229d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.822ex; height:2.676ex;" alt="{\displaystyle r^{2}\pi \,}"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><b>Тетива</b> је дуж која спаја две тачке на кружници. </p><p><b>Централни угао</b> је угао из центра круга под којим се види дата тетива. </p><p><b>Периферни угао</b> је угао из тачке на кружници под којим се види дата тетива. </p><p><b>Тангента</b> је <a href="/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%B0_(%D0%BB%D0%B8%D0%BD%D0%B8%D1%98%D0%B0)" title="Права (линија)">права</a> која додирује кружницу (у једној тачки). </p> <div class="mw-heading mw-heading3"><h3 id="Остале_дефиниције"><span id=".D0.9E.D1.81.D1.82.D0.B0.D0.BB.D0.B5_.D0.B4.D0.B5.D1.84.D0.B8.D0.BD.D0.B8.D1.86.D0.B8.D1.98.D0.B5"></span>Остале дефиниције</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=2" title="Уредите одељак „Остале дефиниције”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=2" title="Уреди извор одељка: Остале дефиниције"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Кружна трансформација</b> равни је трансформација у којој свака кружница или права прелази у кружницу или праву. Кружна трансформација је производ две трансформације: инверзије и сличности. Примери кружних трансформација су: <a href="/wiki/%D0%9A%D1%80%D0%B5%D1%82%D0%B0%D1%9A%D0%B5" title="Кретање">кретање</a>, <a href="/w/index.php?title=%D0%A1%D0%BB%D0%B8%D1%87%D0%BD%D0%BE%D1%81%D1%82&action=edit&redlink=1" class="new" title="Сличност (страница не постоји)">сличност</a>, <a href="/wiki/%D0%98%D0%BD%D0%B2%D0%B5%D1%80%D0%B7%D0%B8%D1%98%D0%B0" title="Инверзија">инверзија</a>. Кружна трансформација је (једно од) <a href="/w/index.php?title=%D0%9A%D0%BE%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%BD%D0%BE_%D0%BF%D1%80%D0%B5%D1%81%D0%BB%D0%B8%D0%BA%D0%B0%D0%B2%D0%B0%D1%9A%D0%B5&action=edit&redlink=1" class="new" title="Конформно пресликавање (страница не постоји)">конформних пресликавања</a>. </p><p><b>Кружни цилиндар</b> (<a href="/wiki/%D0%95%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0" title="Елементарна геометрија">елементарна геометрија</a>) је цилиндар, тј. <a href="/wiki/%D0%92%D0%B0%D1%99%D0%B0%D0%BA_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Ваљак (геометрија)">ваљак</a> чија је <a href="/w/index.php?title=%D0%94%D0%B8%D1%80%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%81%D0%B0&action=edit&redlink=1" class="new" title="Директриса (страница не постоји)">директриса</a> (водиља) кружница. Ако је изводница К. ц. нормална на његову основу, К. ц. се назива прави; ако је пак изводница коса према основи, К. ц. је кос. </p><p>Обично, под појмом кружни цилиндар подразумева се прав кружни цилиндар. Прав кружни цилиндар се може замислити као фигура образована обртањем <a href="/wiki/%D0%A7%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%BE%D1%83%D0%B3%D0%B0%D0%BE" title="Четвороугао">правоугаоника</a> око његове странице. </p><p><b>Кружни конус</b> (у елементарној геометрији) је <a href="/wiki/%D0%9A%D1%83%D0%BF%D0%B0_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Купа (геометрија)">конус</a> (купа) чија је директриса (водиља) кружница. Врх правог кружног конуса се у ортогоналној пројекцији пројектује у центар његове основе. Прав кружни конус се добије обртањем правоуглог троугла око катете. Прав кружни конус се назива једноставно конус. </p><p>Врх косог кружног конуса се у ортогоналној пројекцији не пројектује у центар основе. </p><p>Ако се кружни конус пресече са равни која није <a href="/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%BD%D0%BE%D1%81%D1%82_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Паралелност (геометрија)">паралелна</a> основи, може се у пресеку добити и круг. </p><p><b><a href="/w/index.php?title=%D0%90%D0%BF%D0%BE%D0%BB%D0%BE%D0%BD%D0%B8%D1%98%D0%B5%D0%B2%D0%B0_%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&redlink=1" class="new" title="Аполонијева кружница (страница не постоји)">Аполонијева кружница</a></b> је <a href="/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D1%81%D0%BA%D0%BE_%D0%BC%D0%B5%D1%81%D1%82%D0%BE_%D1%82%D0%B0%D1%87%D0%B0%D0%BA%D0%B0" title="Геометријско место тачака">геометријско место тачака</a> М равни чији је однос одстојања од две дате тачке A и B, које леже у истој овој равни, константна величина <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(k\not =0,k\not =1):AM:BM=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mi>k</mi> <mo>≠</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mi>A</mi> <mi>M</mi> <mo>:</mo> <mi>B</mi> <mi>M</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k(k\not =0,k\not =1):AM:BM=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7954b685c6ef94130dd56ee5005d6a2e803877b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.574ex; height:2.843ex;" alt="{\displaystyle k(k\not =0,k\not =1):AM:BM=k}"></span>. Аполонијева кружница се користи у решавању <a href="/w/index.php?title=%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D1%81%D0%BA%D0%B5_%D0%BA%D0%BE%D0%BD%D1%81%D1%82%D1%80%D1%83%D0%BA%D1%86%D0%B8%D1%98%D0%B5&action=edit&redlink=1" class="new" title="Геометријске конструкције (страница не постоји)">геометријских конструктивних</a> задатака методом геометријских места тачака. На пример: конструкција троугла ако је задата страница, висина на ту страницу и однос остале две странице троугла; страница, њено теме датог троугла и однос остале две странице; када је поред осталих дат однос две висине троугла. Аполонијева кружница је названа по старогрчком научнику <a href="/wiki/%D0%90%D0%BF%D0%BE%D0%BB%D0%BE%D0%BD%D0%B8%D1%98%D0%B5" title="Аполоније">Аполонију</a> из Перга, који ју је изучавао у 3. веку п. н. е.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><b>Кружница девет тачака</b> је кружница на којој леже средине страна <a href="/wiki/%D0%A2%D1%80%D0%BE%D1%83%D0%B3%D0%B0%D0%BE" title="Троугао">троугла</a>, подножја његових висина и средине сегмената висина између темена и <a href="/w/index.php?title=%D0%9E%D1%80%D1%82%D0%BE%D1%86%D0%B5%D0%BD%D1%82%D0%B0%D1%80&action=edit&redlink=1" class="new" title="Ортоцентар (страница не постоји)">ортоцентра</a>. Центар кружнице девет тачака се поклапа са средином дужи која спаја ортоцентар троугла с центром описане кружнице. Полупречник кружнице девет тачака је једнак половини пречника описане кружнице датог троугла. Кружница девет тачака се назива и <a href="/wiki/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B0_%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0" title="Ојлерова кружница">Ојлерова кружница</a>. </p><p><b>Кружница кривине</b> криве у простору у тачки М је кружница која лежи у <a href="/w/index.php?title=%D0%9E%D1%81%D0%BA%D1%83%D0%BB%D0%B0%D1%82%D0%BE%D1%80%D0%BD%D0%B0_%D1%80%D0%B0%D0%B2%D0%B0%D0%BD&action=edit&redlink=1" class="new" title="Оскулаторна раван (страница не постоји)">оскулаторној равни</a> криве у тачки М, чији је радијус једнак <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 {1}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e8560804e21ad155b4f9d8e8a8acd6fae698bb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.047ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{k}}}"></span> где је k <a href="/wiki/Zakrivljenost" title="Zakrivljenost">кривина</a> криве у тачки М, на растојању <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 MO={\frac {1}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>O</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MO={\frac {1}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/528500ce2f1eb3157972562d315f67b3da0e2187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.361ex; height:5.343ex;" alt="{\displaystyle MO={\frac {1}{k}}}"></span>. Кружница кривине не постоји у тачки у којој је кривина криве једнака нули. Кружница кривине има с кривом у тачки М додир чији ред није мањи од 2. Кружница кривине се назива и <a href="/w/index.php?title=%D0%9E%D1%81%D0%BA%D1%83%D0%BB%D0%B0%D1%82%D0%BE%D1%80%D0%BD%D0%B0_%D0%BA%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&redlink=1" class="new" title="Оскулаторна кружница (страница не постоји)">оскулаторна кружница</a>. </p><p><b>Концентричне кружнице</b> су кружнице које имају заједнички центар и леже у истој равни. </p><p><b>Неконцентричне</b> кружнице називају се и <b>ексцентричне</b>. </p><p><b>Конфокалне криве</b> су криве 2. реда (<a href="/wiki/%D0%9A%D0%BE%D0%BD%D1%83%D1%81%D0%BD%D0%B8_%D0%BF%D1%80%D0%B5%D1%81%D0%B5%D0%BA" title="Конусни пресек">конусни пресеци</a>) које имају заједничке жиже (фокусе). </p> <div class="mw-heading mw-heading2"><h2 id="Елементарна_(Еуклидска)_геометрија"><span id=".D0.95.D0.BB.D0.B5.D0.BC.D0.B5.D0.BD.D1.82.D0.B0.D1.80.D0.BD.D0.B0_.28.D0.95.D1.83.D0.BA.D0.BB.D0.B8.D0.B4.D1.81.D0.BA.D0.B0.29_.D0.B3.D0.B5.D0.BE.D0.BC.D0.B5.D1.82.D1.80.D0.B8.D1.98.D0.B0"></span>Елементарна (Еуклидска) геометрија</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=3" title="Уредите одељак „Елементарна (Еуклидска) геометрија”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=3" title="Уреди извор одељка: Елементарна (Еуклидска) геометрија"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>1. Теорема</dt> <dd>Централни угао је двоструко већи од периферног над истом тетивом<b>. <br /></b></dd></dl> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/%D0%94%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Centralni-ugao.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b1/Centralni-ugao.gif" decoding="async" width="210" height="218" class="mw-file-element" data-file-width="210" data-file-height="218" /></a></span> <br /> </p> <dl><dt>Доказ</dt> <dd>Дата су кружница k тетива <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\in k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f95929a36b31bbb12f5b98a6a8b88411eba5e464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.593ex; height:2.509ex;" alt="{\displaystyle A,B\in k}"></span> централни угао <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 ACB=\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>C</mi> <mi>B</mi> <mo>=</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ACB=\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ef9db7a78346ec20939eb75ade2fbe4ec35e8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.462ex; height:2.176ex;" alt="{\displaystyle ACB=\theta }"></span> и периферни угао <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 APB=\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>P</mi> <mi>B</mi> <mo>=</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle APB=\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45fac9a0b33c44f9f3b56ad110dcef5ec65327da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.736ex; height:2.509ex;" alt="{\displaystyle APB=\phi }"></span>.</dd></dl> <dl><dd>Дужи CA, CB и CP су једнаке (полупречници), па је троугао BCP једнакокраки. Исто тако и троугао ACP је једнакокраки. PP' је пречник круга, а AP’ и BP’ су такође тетиве.</dd></dl> <dl><dd>Спољашњи угао троугла једнак је збиру два унутрашња њему несуседна угла, тј. <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 \theta _{1}=2\phi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{1}=2\phi _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fc67fe34fe964afe66af048f8e5d22b5b6b153e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.845ex; height:2.509ex;" alt="{\displaystyle \theta _{1}=2\phi _{1}}"></span>, и отуда:</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\theta _{1}+\theta _{2}=2\phi _{1}+2\phi _{2}=2\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\theta _{1}+\theta _{2}=2\phi _{1}+2\phi _{2}=2\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08b8ab6a06d17094f616ab332c0c46fd0592f8da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.108ex; height:2.509ex;" alt="{\displaystyle \theta =\theta _{1}+\theta _{2}=2\phi _{1}+2\phi _{2}=2\phi }"></span>.</dd></dl> <dl><dt>2. Теорема</dt> <dd>Периферни угао над пречником је прав.</dd></dl> <dl><dt>Доказ</dt> <dd>Из претходне теореме, јер је централни угао над пречником 180°, а пола од тога је прави угао.</dd></dl> <dl><dt>3. Теорема</dt> <dd>Угао између тетиве и тангенте повучених из исте тачке кружнице једнак је периферном над том тетивом.</dd></dl> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/%D0%94%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Ugao-tangente.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/9a/Ugao-tangente.gif" decoding="async" width="166" height="170" class="mw-file-element" data-file-width="166" data-file-height="170" /></a></span> </p> <dl><dt>Доказ</dt> <dd>Дат су круг k, тангента t и тетива AB. AP је пречник круга па је угао у B прав. Углови BAt и APB имају окомите краке, тј. једнаки су!</dd></dl> <dl><dt>4. Теорема</dt> <dd>Периферни углови над истом тетивом једнаки су или су суплементни. Ако су са различитих страна тетиве они су суплементни.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Растојање_тачке_од_кружнице"><span id=".D0.A0.D0.B0.D1.81.D1.82.D0.BE.D1.98.D0.B0.D1.9A.D0.B5_.D1.82.D0.B0.D1.87.D0.BA.D0.B5_.D0.BE.D0.B4_.D0.BA.D1.80.D1.83.D0.B6.D0.BD.D0.B8.D1.86.D0.B5"></span>Растојање тачке од кружнице</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=4" title="Уредите одељак „Растојање тачке од кружнице”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=4" title="Уреди извор одељка: Растојање тачке од кружнице"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Дефиниција 1</dt></dl> <p>Скуп свих тачака равни чија је удаљеност од дате тачке О те равни једнака датој дужи називамо кружница с центром у О и полупречником (радијусом) r. </p><p>Спојимо ли се тачка C са тачкама кружнице K(O,r) добија се бесконачан скуп дужи за C ≠ O. У случају C = O то је нулта дуж. Поставља се питање постојања у овом скупу дужи од које ни једна дуж скупа није мања, и такве дужи која није мања ни од једне дужи скупа. То су дужи CA и CB, где су A, B тачке кружнице које леже на централној правој која пролази кроз C. Тачка A је са оне стране тачке O са које је C, а B је са супротне стране. </p> <dl><dt>Дефиниција 2</dt> <dd></dd></dl> <p>Element m скупа E (у коме између елемената постоји <a href="/wiki/%D0%A0%D0%B5%D0%BB%D0%B0%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Релација (математика)">релација</a> < или > ) који није већи ни од једног елемента <a href="/wiki/%D0%A1%D0%BA%D1%83%D0%BF" title="Скуп">скупа</a> назива се минимум (најмањи елемент скупа E). Елемент који није мањи ни од једног елемента скупа је максимум (највећи) елемент скупа E. </p><p>У наведеном случају дужи AB и AC су минимум и максимуму у скупу дужи. </p> <dl><dt>Дефиниција 3</dt> <dd></dd></dl> <p>Минимум скупа растојања дате тачке од скупа назива се растојање те тачке од скупа. </p><p><b>Теорема 1</b> </p><p>Нека је дата тачка C и кружница K(O,r) и при том C ≠ O и нека су тачке A, B тачке кружнице које леже на централној правој, која пролази тачком C. Тачка A нека је с оне стране с које је тачка О, а B са супротне стране од О. Тада од свих тачака кружнице тачка A има најмање, а тачка B највеће растојање од C и при томе је </p> <dl><dd>CA = │CO - r│ i CB = CO + r</dd></dl> <p>Бесконачни скупови не морају да имају минимум и максимум. </p><p>Пример </p><p>Скуп бројева 1,1/2, ¼, 1/8,...има максимум, а нема минимум </p> <div class="mw-heading mw-heading2"><h2 id="Заједничке_тачке_кружница"><span id=".D0.97.D0.B0.D1.98.D0.B5.D0.B4.D0.BD.D0.B8.D1.87.D0.BA.D0.B5_.D1.82.D0.B0.D1.87.D0.BA.D0.B5_.D0.BA.D1.80.D1.83.D0.B6.D0.BD.D0.B8.D1.86.D0.B0"></span>Заједничке тачке кружница</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=5" title="Уредите одељак „Заједничке тачке кружница”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=5" title="Уреди извор одељка: Заједничке тачке кружница"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Нека су задане две кружнице K(C, R) и k(O,r). Одредимо међусобни положај ових кружница. Повучемо ли централну праву CO ових кружница, са A, B означимо тачке друге кружнице и то са А ону која лежи са оне стране од тачке О са које је тачка C, а са B тачку друге кружнице. </p><p>Посматрајмо дужи R – r, CO и R + r за R > r Између ових дужи постоји један и само један од ових односа </p> <ol><li><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 CO>R+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>O</mi> <mo>></mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CO>R+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27124b47de2feaa5154f43153cf2724f1816f19a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.291ex; height:2.343ex;" alt="{\displaystyle CO>R+r}"></span></li> <li><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 CO=R+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>O</mi> <mo>=</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CO=R+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf8771d6a52136e393babddf5b4e978f583c687e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.291ex; height:2.343ex;" alt="{\displaystyle CO=R+r}"></span></li> <li><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-r<CO<R+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>−</mo> <mi>r</mi> <mo><</mo> <mi>C</mi> <mi>O</mi> <mo><</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R-r<CO<R+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bff885192ca4e6904c26f8ffdf139fa7a231a765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.043ex; height:2.343ex;" alt="{\displaystyle R-r<CO<R+r}"></span></li> <li><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 CO<R-r(R>r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>O</mi> <mo><</mo> <mi>R</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>></mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CO<R-r(R>r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f59071aa3effea61e94d3300bc3ea8b965216fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.012ex; height:2.843ex;" alt="{\displaystyle CO<R-r(R>r)}"></span></li> <li><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 CO=R-r(R>r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>O</mi> <mo>=</mo> <mi>R</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>></mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CO=R-r(R>r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e7b3e28f7b53ebc44ec85feb060f1fd59666fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.012ex; height:2.843ex;" alt="{\displaystyle CO=R-r(R>r)}"></span></li></ol> <div class="mw-heading mw-heading3"><h3 id="Пресек_кружница_празан_скуп"><span id=".D0.9F.D1.80.D0.B5.D1.81.D0.B5.D0.BA_.D0.BA.D1.80.D1.83.D0.B6.D0.BD.D0.B8.D1.86.D0.B0_.D0.BF.D1.80.D0.B0.D0.B7.D0.B0.D0.BD_.D1.81.D0.BA.D1.83.D0.BF"></span>Пресек кружница празан скуп</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=6" title="Уредите одељак „Пресек кружница празан скуп”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=6" title="Уреди извор одељка: Пресек кружница празан скуп"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>За <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 CO>R+r<=>CO-r>R<=>CA>R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>O</mi> <mo>></mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mo><=></mo> <mi>C</mi> <mi>O</mi> <mo>−</mo> <mi>r</mi> <mo>></mo> <mi>R</mi> <mo><=></mo> <mi>C</mi> <mi>A</mi> <mo>></mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CO>R+r<=>CO-r>R<=>CA>R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a4ee97c263a675f88b94f175839d703d678e9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:46.384ex; height:2.343ex;" alt="{\displaystyle CO>R+r<=>CO-r>R<=>CA>R}"></span></li></ul> <p>Све тачке једне кружнице су изван друге кружнице. </p> <ul><li><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 O<R-r<=>CO-r<R<=>CB<R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo><</mo> <mi>R</mi> <mo>−</mo> <mi>r</mi> <mo><=></mo> <mi>C</mi> <mi>O</mi> <mo>−</mo> <mi>r</mi> <mo><</mo> <mi>R</mi> <mo><=></mo> <mi>C</mi> <mi>B</mi> <mo><</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O<R-r<=>CO-r<R<=>CB<R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01783d45d7ab30498e2ce7bedc0b78fdad724fc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:44.638ex; height:2.343ex;" alt="{\displaystyle O<R-r<=>CO-r<R<=>CB<R}"></span></li></ul> <p>Све тачке једне кружнице су унутар друге кружнице. </p> <div class="mw-heading mw-heading2"><h2 id="Тангента_кружнице"><span id=".D0.A2.D0.B0.D0.BD.D0.B3.D0.B5.D0.BD.D1.82.D0.B0_.D0.BA.D1.80.D1.83.D0.B6.D0.BD.D0.B8.D1.86.D0.B5"></span>Тангента кружнице</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=7" title="Уредите одељак „Тангента кружнице”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=7" title="Уреди извор одељка: Тангента кружнице"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Тангента кружнице са средиштем <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(0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3932c29e601bde66dad3e5efc6ad768cdac9d155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.667ex; height:2.843ex;" alt="{\displaystyle S(0,0)}"></span></dt></dl> <p>Тангента кружнице која има средиште у <a href="/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B8_%D0%BF%D0%BE%D1%87%D0%B5%D1%82%D0%B0%D0%BA" title="Координатни почетак">координатном почетку</a> координатног система и која пролази точком <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e29c26c8c2d9f87c652c124f580fa4626bb6944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.057ex; height:2.843ex;" alt="{\displaystyle T(x_{0},y_{0})}"></span> </p><p>на кружници, одређена је координатама точке Т и коефицијентом смера тангенте. Диференцирањем једначине кружнице налази се да је: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {2xdx+2ydy}={0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>x</mi> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mi>d</mi> <mi>y</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {2xdx+2ydy}={0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325c8c7bf1905ee423cba9e25b20ff50e1079acf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.215ex; height:2.509ex;" alt="{\displaystyle {2xdx+2ydy}={0}\,}"></span></dd></dl> <p>одакле следи да је </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'={\frac {dy}{dx}}=\tan \alpha \,=-{\frac {x_{0}}{y_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mspace width="thinmathspace" /> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'={\frac {dy}{dx}}=\tan \alpha \,=-{\frac {x_{0}}{y_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ec0f5b0b39a27a5e626d424bb094c769202616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.172ex; height:5.843ex;" alt="{\displaystyle y'={\frac {dy}{dx}}=\tan \alpha \,=-{\frac {x_{0}}{y_{0}}}}"></span></dd></dl> <p>једначина тангенте на кружницу </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y-y_{0}=-{{\frac {x_{0}}{y_{0}}}(x-x_{0})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y-y_{0}=-{{\frac {x_{0}}{y_{0}}}(x-x_{0})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94576bab528332b96df0005abe616b22e9b98fe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.679ex; height:5.176ex;" alt="{\displaystyle y-y_{0}=-{{\frac {x_{0}}{y_{0}}}(x-x_{0})}}"></span></dd></dl> <p>одакле се сређивањем налази и други облик једначине тангенте кружнице </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}x+y_{0}y=r^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>y</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}x+y_{0}y=r^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea6f97c4247862947205161733f2eb406303727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.491ex; height:3.009ex;" alt="{\displaystyle x_{0}x+y_{0}y=r^{2}\,}"></span>.</dd></dl> <dl><dt>Тангента кружнице са средиштем у <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(p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9f0a69ad929e6f887e8a18ddabc0f15b90987e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.581ex; height:2.843ex;" alt="{\displaystyle S(p,q)}"></span></dt></dl> <p>Тангента кружнице која има средиште у тачки <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(p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9f0a69ad929e6f887e8a18ddabc0f15b90987e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.581ex; height:2.843ex;" alt="{\displaystyle S(p,q)}"></span> и која пролази тачком <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e29c26c8c2d9f87c652c124f580fa4626bb6944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.057ex; height:2.843ex;" alt="{\displaystyle T(x_{0},y_{0})}"></span> на кружници одређена је координатама тачке Т и коефицијентом смера тангенте. Диференцирањем једначине кружнице налази се да је: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {2(x-p)dx+2(y-q)dy}={0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>y</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {2(x-p)dx+2(y-q)dy}={0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3a5b63852d886fd0e41707f16edaed363fa184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.754ex; height:2.843ex;" alt="{\displaystyle {2(x-p)dx+2(y-q)dy}={0}\,}"></span></dd></dl> <p>одакле следи да је </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'={\frac {dy}{dx}}=\tan \alpha \,=-{\frac {x_{0}-p}{y_{0}-q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mspace width="thinmathspace" /> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>p</mi> </mrow> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>q</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'={\frac {dy}{dx}}=\tan \alpha \,=-{\frac {x_{0}-p}{y_{0}-q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e77be73ba89fb11c04cafa041bbc32b49dba9b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.182ex; height:5.843ex;" alt="{\displaystyle y'={\frac {dy}{dx}}=\tan \alpha \,=-{\frac {x_{0}-p}{y_{0}-q}}}"></span></dd></dl> <p>те се сличним поступком налази да је једначина тангенте кружнице </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y-y_{0}=-{{\frac {x_{0}-p}{y_{0}-q}}(x-x_{0})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>p</mi> </mrow> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>q</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y-y_{0}=-{{\frac {x_{0}-p}{y_{0}-q}}(x-x_{0})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/747d22378062eda7749e0731bc5f2fcd54da74a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.689ex; height:5.676ex;" alt="{\displaystyle y-y_{0}=-{{\frac {x_{0}-p}{y_{0}-q}}(x-x_{0})}}"></span></dd></dl> <p>одакле се сређивањем налази и други облик једначине тангенте кружнице </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0}-p)(x-p)+(y_{0}-q)(y-q)=r^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0}-p)(x-p)+(y_{0}-q)(y-q)=r^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/969ed3a03a6892a024f2c7b5f47ad0aee5262049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.568ex; height:3.176ex;" alt="{\displaystyle (x_{0}-p)(x-p)+(y_{0}-q)(y-q)=r^{2}\,}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Тангирање_кружница"><span id=".D0.A2.D0.B0.D0.BD.D0.B3.D0.B8.D1.80.D0.B0.D1.9A.D0.B5_.D0.BA.D1.80.D1.83.D0.B6.D0.BD.D0.B8.D1.86.D0.B0"></span>Тангирање кружница</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=8" title="Уредите одељак „Тангирање кружница”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=8" title="Уреди извор одељка: Тангирање кружница"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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 CO=R+r<=>CO-r<R<=>CA=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>O</mi> <mo>=</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mo><=></mo> <mi>C</mi> <mi>O</mi> <mo>−</mo> <mi>r</mi> <mo><</mo> <mi>R</mi> <mo><=></mo> <mi>C</mi> <mi>A</mi> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CO=R+r<=>CO-r<R<=>CA=R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80f7e4f9b6049c21fb963f9febbc34a2a2221b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:46.384ex; height:2.343ex;" alt="{\displaystyle CO=R+r<=>CO-r<R<=>CA=R}"></span></li></ul> <p>Тачка А друге кружнице припада тачкама прве кружнице. Све остале тачке су изван прве кружнице. За кружнице које имају једну и само једну заједничку тачку и она лежи на правој CO кажемо да се оне додирују споља у тачки A. </p> <ul><li><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 CO=R-r(R>r)<=>CO-r=R<=>CB=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>O</mi> <mo>=</mo> <mi>R</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>></mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo><=></mo> <mi>C</mi> <mi>O</mi> <mo>−</mo> <mi>r</mi> <mo>=</mo> <mi>R</mi> <mo><=></mo> <mi>C</mi> <mi>B</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CO=R-r(R>r)<=>CO-r=R<=>CB=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff392a62aeb074476458834b621a0b5015e360e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.41ex; height:2.843ex;" alt="{\displaystyle CO=R-r(R>r)<=>CO-r=R<=>CB=r}"></span></li></ul> <p>Тачка B припада првој кружници све остале тачке друге кружнице су унутар прве кружнице. Ако две кружнице имају дијаметрално распоређене две заједничке тачке M на правој CO онда су оне дијаметрално супротне за сваку од те две тачке које леже на правој. За сваку од те две кружнице па се оне поклапају. </p> <div class="mw-heading mw-heading3"><h3 id="Пресек_кружница"><span id=".D0.9F.D1.80.D0.B5.D1.81.D0.B5.D0.BA_.D0.BA.D1.80.D1.83.D0.B6.D0.BD.D0.B8.D1.86.D0.B0"></span>Пресек кружница</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=9" title="Уредите одељак „Пресек кружница”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=9" title="Уреди извор одељка: Пресек кружница"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>R – r < CO < R + r ( R < r) </p> <ul><li>A је у, B изван K(C,R)</li> <li>R – r < CO => CB > R</li></ul> <p>B је ван K (C,R) </p> <dl><dd>CO < R + r => CA < RA не у кружници.</dd></dl> <p>Од две дијаметрално распоређене тачке једна је у, а друга ван кружнице. Тачке A, B деле кружницу на два дела. </p><p><b>Аксиом 2</b> </p><p>Ако се један крај лука налази у кружници, а други изван је онда тај лук са кружницом има једну и само једну заједничку тачку. </p><p><b>Теорема 2</b> </p><p>Заједничка тачка две кружнице које се додирују лежи на њиховој заједничкој централној правој, и обратно две различите кружнице које имају заједничку тачку на централној правој се додирују. Ако две кружнице имају заједничку тачку која не лежи на централној правој, имају још једну заједничку тачку. </p><p><b>Теорема 3</b> </p><p>Две кружнице K(C,R) и k(O,r) </p> <ul><li>одакле се сређивање налази и други облик једначине тангенте кружнице <ul><li>CO > R + r (свака од кружница је изван друге кружнице)</li> <li>CO < R - r (кружница мањег пречника је унутар кружнице већег пречника)</li></ul></li> <li>Имају једну и само једну заједничку тачку која лежи на заједничкој централној правој <ul><li>CO = R + r све тачке кружнице осим заједничке су изван друге кружнице</li></ul></li> <li>R – r < CO < R + r имају две и само две заједничке тачке које леже са разних страна централне праве.</li></ul> <p><b>Теорема 4</b> </p><p>Да би две кружнице имале заједничких тачака у случају да се центар прве кружнице налази </p> <ol><li>на другој кружници</li> <li>у другој кружници</li></ol> <p>потребно је и довољно да буде </p> <ol><li>R ≤ 2r</li> <li>CA < R < CB</li></ol> <p>где су CA и CB одсечци на које центар О дели дијаметар AB кружнице k(O, r). </p> <div class="mw-heading mw-heading2"><h2 id="Полара_кружнице"><span id=".D0.9F.D0.BE.D0.BB.D0.B0.D1.80.D0.B0_.D0.BA.D1.80.D1.83.D0.B6.D0.BD.D0.B8.D1.86.D0.B5"></span>Полара кружнице</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=10" title="Уредите одељак „Полара кружнице”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=10" title="Уреди извор одељка: Полара кружнице"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Конјуговане тачке у односу на кружницу </p><p>Тачке P и P<sub>1</sub> су конјуговане у односу на кружницу ако задовољавају формулу </p><p><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 {\overrightarrow {OM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>M</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {OM}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a35bb30cf6e3998acb8ceeef6b2d23696a359e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.4ex; width:4.468ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {OM}}}"></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 {\overrightarrow {ON}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>N</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {ON}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6369660e0b4430a20703805a10b9a464be3be61d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.4ex; width:4.095ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {ON}}}"></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 R^{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce07e278be3e058a6303de8359f8b4a4288264a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}"></span> </p><p>Ово је једначина <b>поларе кружнице</b>. Скуп коњугованих тачака кружнице је права. </p> <ol><li>Полара сече кружницу ако је тачка M ван крижнице.</li> <li>тангента је кружнице ако је M на кружници</li> <li>Нема заједничких тачака ако је M у кружници</li> <li>Пролази кроз центар кружнице ако је M у бесконачности</li> <li>Ако је <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 O\equiv M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo>≡</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O\equiv M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d08ae557006b3bdecefaa7f90c771119c66aa968" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.314ex; height:2.176ex;" alt="{\displaystyle O\equiv M}"></span> онда је полара у бесконачности.</li></ol> <p><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 {\overrightarrow {OM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>M</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {OM}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a35bb30cf6e3998acb8ceeef6b2d23696a359e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.4ex; width:4.468ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {OM}}}"></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 {\overrightarrow {OP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {OP}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff45f0fefb0bfc3dd4aa31b8bc68ea0bfcd39fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.4ex; width:3.813ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {OP}}}"></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 R^{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce07e278be3e058a6303de8359f8b4a4288264a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}"></span> </p><p><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 OA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db864cf94e2655d6a7b56c7479f63933e97afb06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.516ex; height:2.176ex;" alt="{\displaystyle OA}"></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 OB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d361e055353ea627ef7fb6835634db09826482cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.537ex; height:2.176ex;" alt="{\displaystyle OB}"></span> </p><p><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 OA^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OA^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74d0a8c67f1a5633de96e9d2ae19360fd619c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.571ex; height:2.676ex;" alt="{\displaystyle OA^{2}}"></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 OB^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OB^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3ec71651918b2bca2fd196775688f57be20480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.592ex; height:2.676ex;" alt="{\displaystyle OB^{2}}"></span> </p><p>(<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 OA^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OA^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74d0a8c67f1a5633de96e9d2ae19360fd619c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.571ex; height:2.676ex;" alt="{\displaystyle OA^{2}}"></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 OB^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OB^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3ec71651918b2bca2fd196775688f57be20480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.592ex; height:2.676ex;" alt="{\displaystyle OB^{2}}"></span>)=0 </p><p>(<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 AO}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AO}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e32f2a93c315670edb3f60f7cc07f59cf38e7260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.516ex; height:2.176ex;" alt="{\displaystyle AO}"></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 OM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498f054a4449d57c02268bf851edb3a3b2b61913" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.215ex; height:2.176ex;" alt="{\displaystyle OM}"></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 BO}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BO}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56842b7455364a817a18dcb512dac808494ece5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.537ex; height:2.176ex;" alt="{\displaystyle BO}"></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 OM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498f054a4449d57c02268bf851edb3a3b2b61913" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.215ex; height:2.176ex;" alt="{\displaystyle OM}"></span>)=0 </p><p><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 AM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14612d61c481c5020a2205c9fcf2b3ffb44bdb79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.185ex; height:2.176ex;" alt="{\displaystyle AM}"></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 BM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5259a0a427188429b545c0f67d10209af65962e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.206ex; height:2.176ex;" alt="{\displaystyle BM}"></span>=0 </p> <div class="mw-heading mw-heading3"><h3 id="Аполонијева_кружница"><span id=".D0.90.D0.BF.D0.BE.D0.BB.D0.BE.D0.BD.D0.B8.D1.98.D0.B5.D0.B2.D0.B0_.D0.BA.D1.80.D1.83.D0.B6.D0.BD.D0.B8.D1.86.D0.B0"></span>Аполонијева кружница</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=11" title="Уредите одељак „Аполонијева кружница”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=11" title="Уреди извор одељка: Аполонијева кружница"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Геометријско место тачака равни које имају особину да је однос удаљености тих тачака сталан број је кружница – Аполонијева кружница </p><p><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 {MA}{MB}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <mi>A</mi> </mrow> <mrow> <mi>M</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {MA}{MB}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1673bb7f861fdc571a6916c32d2f63a4ee13439d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.496ex; height:5.343ex;" alt="{\displaystyle {\frac {MA}{MB}}=}"></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 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/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></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 \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></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 MA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6250feb34ae98d9b710a8b069d15811718a9c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.185ex; height:2.176ex;" alt="{\displaystyle MA}"></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 kMB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>M</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kMB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4b04d97e10edbca6ce37014fd9b5fd68310321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.417ex; height:2.176ex;" alt="{\displaystyle kMB}"></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 \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></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 MA^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MA^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389e8ed7d21fdf83baf1bb98d05a60ff45a440fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.24ex; height:2.676ex;" alt="{\displaystyle MA^{2}}"></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 k^{2}MB^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>M</mi> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{2}MB^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764534a0627b41a336978bc1cc005beea4131dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.526ex; height:2.676ex;" alt="{\displaystyle k^{2}MB^{2}}"></span> </p><p><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 (MA+kMB)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mi>A</mi> <mo>+</mo> <mi>k</mi> <mi>M</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (MA+kMB)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02eddbd027a5db94c50a435b18a2310731c7d0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.252ex; height:2.843ex;" alt="{\displaystyle (MA+kMB)}"></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 (MA-kMB)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mi>A</mi> <mo>−</mo> <mi>k</mi> <mi>M</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (MA-kMB)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29634a56e37a20ca9ace1479d6025a778db9b792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.252ex; height:2.843ex;" alt="{\displaystyle (MA-kMB)}"></span>=0 </p><p><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 {\dfrac {MA-kMB}{1-k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mi>A</mi> <mo>−</mo> <mi>k</mi> <mi>M</mi> <mi>B</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {MA-kMB}{1-k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d3e59fabc288c6b3920252050df428dba35f18f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.279ex; height:5.676ex;" alt="{\displaystyle {\dfrac {MA-kMB}{1-k}}}"></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 {\dfrac {MA+kMB}{1+k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mi>A</mi> <mo>+</mo> <mi>k</mi> <mi>M</mi> <mi>B</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {MA+kMB}{1+k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75264a3b542a6b33c578cbf010fe5c0627377769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.279ex; height:5.676ex;" alt="{\displaystyle {\dfrac {MA+kMB}{1+k}}}"></span> =0 </p><p>За њеном </p><p><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 {\dfrac {MA-kMB}{1-k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mi>A</mi> <mo>−</mo> <mi>k</mi> <mi>M</mi> <mi>B</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {MA-kMB}{1-k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d3e59fabc288c6b3920252050df428dba35f18f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.279ex; height:5.676ex;" alt="{\displaystyle {\dfrac {MA-kMB}{1-k}}}"></span> са <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 MC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c34f200ee412708546670f54ce4db2e53b8e15f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.209ex; height:2.176ex;" alt="{\displaystyle MC}"></span> и </p><p><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 {\dfrac {MA+kMB}{1+k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mi>A</mi> <mo>+</mo> <mi>k</mi> <mi>M</mi> <mi>B</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {MA+kMB}{1+k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75264a3b542a6b33c578cbf010fe5c0627377769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.279ex; height:5.676ex;" alt="{\displaystyle {\dfrac {MA+kMB}{1+k}}}"></span> са <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 MD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea9cad6042d6f196d8bdfc83ea215262c37923d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.366ex; height:2.176ex;" alt="{\displaystyle MD}"></span> </p><p>Имамо <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 MC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c34f200ee412708546670f54ce4db2e53b8e15f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.209ex; height:2.176ex;" alt="{\displaystyle MC}"></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 MD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea9cad6042d6f196d8bdfc83ea215262c37923d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.366ex; height:2.176ex;" alt="{\displaystyle MD}"></span>=0 кружница са пречником CD. </p> <div class="mw-heading mw-heading2"><h2 id="Кружнице_у_-{p}--нормама_и_бројеви_'"`UNIQ--postMath-0000004D-QINU`"'"><span id=".D0.9A.D1.80.D1.83.D0.B6.D0.BD.D0.B8.D1.86.D0.B5_.D1.83_-.7Bp.7D--.D0.BD.D0.BE.D1.80.D0.BC.D0.B0.D0.BC.D0.B0_.D0.B8_.D0.B1.D1.80.D0.BE.D1.98.D0.B5.D0.B2.D0.B8_.7F.27.22.60UNIQ--postMath-0000004D-QINU.60.22.27.7F"></span>Кружнице у p-нормама и бројеви <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 \pi _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37668045f8b8d371f2e8f0d98cba26c9e7fac7d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:2.343ex;" alt="{\displaystyle \pi _{p}}"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=12" title="Уредите одељак „Кружнице у p-нормама и бројеви '"`UNIQ--postMath-0000004D-QINU`"'”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=12" title="Уреди извор одељка: Кружнице у p-нормама и бројеви '"`UNIQ--postMath-0000004D-QINU`"'"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Досад је удаљеност рачуната помоћу метрике <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9276f8f68c5c23329de74ad76e69f6801358fb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{2}}"></span>. За дефинисање појма кружнице се може уместо метрике <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9276f8f68c5c23329de74ad76e69f6801358fb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{2}}"></span> узети нека друга метрика d. </p><p>Скуп </p><p><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={(x,y)\in R^{2}:d((x,y),(x_{0},y_{0}))=r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={(x,y)\in R^{2}:d((x,y),(x_{0},y_{0}))=r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ff3bbb6a17cf3feaf1598ac98ab0569968ca86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.477ex; height:3.176ex;" alt="{\displaystyle S={(x,y)\in R^{2}:d((x,y),(x_{0},y_{0}))=r}}"></span> </p><p>представља кружницу радијуса r са средиштем у (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0},y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75dfe041b2201f4d63e04777fcfd274f271bdec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.611ex; height:2.009ex;" alt="{\displaystyle x_{0},y_{0}}"></span>) с обзиром на метрику d. </p><p>Кружница радијуса r са средиштем у координатном почетку с обзиром на <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_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a3efb685b5e6d0fa5be00ba01cf4e30a5641ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.268ex; height:2.843ex;" alt="{\displaystyle d_{p}}"></span> је скуп </p><p><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_{p}={(x,y)\in R^{2}:x^{p}+y^{p}=r^{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{p}={(x,y)\in R^{2}:x^{p}+y^{p}=r^{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ab776ac310fb55fde5dadedaac1a658a01b9c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.162ex; height:3.343ex;" alt="{\displaystyle S_{p}={(x,y)\in R^{2}:x^{p}+y^{p}=r^{p}}}"></span> за <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 p\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93c1353080a21b276e79058d82c19c40310653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\geq 1}"></span> </p><p>На овој слици приказане су кружнице <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_{1},S_{2}iS_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>i</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1},S_{2}iS_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcdb62f4b70ce37e7d1d4fdabfefd1a225abd2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.095ex; height:2.509ex;" alt="{\displaystyle S_{1},S_{2}iS_{\infty }}"></span> </p><p>Када би се нацртале и остале кружнице <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_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753062ca2b97967794cc23e3e553f46898493d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.484ex; height:2.843ex;" alt="{\displaystyle S_{p}}"></span>, све би оне биле смештене између <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebfe462c3a7f2183faf705c10fc311580d11c49d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.3ex; height:2.509ex;" alt="{\displaystyle S_{\infty }}"></span>, и што би p био већи, то би кружница <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_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753062ca2b97967794cc23e3e553f46898493d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.484ex; height:2.843ex;" alt="{\displaystyle S_{p}}"></span> била ближе кружнице <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_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebfe462c3a7f2183faf705c10fc311580d11c49d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.3ex; height:2.509ex;" alt="{\displaystyle S_{\infty }}"></span>. </p><p>То је јасно из теорема за максималну норму. </p><p>Узмимо <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6584ba3b7843583b757896c2f0686efc0489e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}"></span>. Нека је </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in R^{2}(x,y)\neq (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in R^{2}(x,y)\neq (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb20c794458ea2ecbaf26fef0dc1964ab704ce0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.582ex; height:3.176ex;" alt="{\displaystyle (x,y)\in R^{2}(x,y)\neq (0,0)}"></span> </p><p>Тада тачка </p><p><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 {1}{\begin{Vmatrix}(x,y)\end{Vmatrix}}}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\begin{Vmatrix}(x,y)\end{Vmatrix}}}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92025c43b4fea1a9cea764b89087641b592fe1c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.569ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\begin{Vmatrix}(x,y)\end{Vmatrix}}}(x,y)}"></span> лежи на кружници <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></span>, јер је </p><p><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 {\begin{Vmatrix}{\frac {1}{{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}}}(x,y)\end{Vmatrix}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Vmatrix}{\frac {1}{{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}}}(x,y)\end{Vmatrix}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a784c41167df8890cfb44c7f40d27d8ed63927e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:20.223ex; height:4.176ex;" alt="{\displaystyle {\begin{Vmatrix}{\frac {1}{{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}}}(x,y)\end{Vmatrix}}=1}"></span> </p><p>Са слике се виде да кружница <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></span> лежи унутар кружнице <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}"></span> па је тачка </p><p><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 {1}{\begin{Vmatrix}(x,y)\end{Vmatrix}}}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\begin{Vmatrix}(x,y)\end{Vmatrix}}}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92025c43b4fea1a9cea764b89087641b592fe1c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.569ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\begin{Vmatrix}(x,y)\end{Vmatrix}}}(x,y)}"></span> унутар кружнице <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}"></span> тј </p><p><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 {\begin{Vmatrix}{\frac {1}{{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}}}(x,y)\end{Vmatrix}}\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Vmatrix}{\frac {1}{{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}}}(x,y)\end{Vmatrix}}\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78f3d51d222e927b1859845ed72a8021de379200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:20.223ex; height:4.176ex;" alt="{\displaystyle {\begin{Vmatrix}{\frac {1}{{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}}}(x,y)\end{Vmatrix}}\leq 1}"></span> </p><p>вреди </p><p><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 {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec7b9575ea70bf8c9b72c0a849c4c61bbdfab78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.017ex; height:3.009ex;" alt="{\displaystyle {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}}"></span> </p><p>Када би се нацртала кружницу радијуса <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> у односу на метрику <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span> односно <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 {\sqrt {2}}S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78f6762bd305d07773c961f016e6eb353511af50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.577ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}S_{1}}"></span> кружница <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}"></span> би била смештена унутар ње. </p><p><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 {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}\leq {\sqrt {2}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}\leq {\sqrt {2}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650d71cd52fe31abbbf0de25b0c690d592be5847" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.115ex; height:3.343ex;" alt="{\displaystyle {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}\leq {\sqrt {2}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}}"></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 {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}\leq {\sqrt {2}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}\leq {\sqrt {2}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3efbbcd96e9fc3c915a9606a7ec6a1dbaf1ae90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.672ex; height:3.343ex;" alt="{\displaystyle {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{1}\leq {\sqrt {2}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{2}}"></span> </p><p>Пропозиција </p><p>За све <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 p,q\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bb8c3eb398868e60013d6c37e07cc3fb397172" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.623ex; height:2.509ex;" alt="{\displaystyle p,q\geq 1}"></span> </p><p><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 {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{\infty }\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{p}\leq {\sqrt[{p}]{\sqrt {2}}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msqrt> <mn>2</mn> </msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{\infty }\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{p}\leq {\sqrt[{p}]{\sqrt {2}}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df635a292745ab518738faf46e547fb231250b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:41.643ex; height:4.843ex;" alt="{\displaystyle {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{\infty }\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{p}\leq {\sqrt[{p}]{\sqrt {2}}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{\infty }}"></span> </p><p><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 {\sqrt[{p}]{^{-1}}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{q}\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{p}\leq {\sqrt[{p}]{2}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{p}]{^{-1}}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{q}\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{p}\leq {\sqrt[{p}]{2}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b4698a72e80769c8a35186bc7112256bbcfd44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:42.202ex; height:4.176ex;" alt="{\displaystyle {\sqrt[{p}]{^{-1}}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{q}\leq {\begin{Vmatrix}(x,y)\end{Vmatrix}}_{p}\leq {\sqrt[{p}]{2}}{\begin{Vmatrix}(x,y)\end{Vmatrix}}_{q}}"></span> </p><p>Геометријски облик кружнице зависи од одабране метрике. </p><p>Израчунајмо обим <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 O_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7898d8340db13ac558a1d8e14f07f3a3b251bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.832ex; height:2.843ex;" alt="{\displaystyle O_{p}}"></span> кружнице <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_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753062ca2b97967794cc23e3e553f46898493d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.484ex; height:2.843ex;" alt="{\displaystyle S_{p}}"></span>. </p><p>Обим кружнице <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}"></span> је <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 O_{2}=2r\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>r</mi> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{2}=2r\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2271b89d8039660937389e667f05d7273535a2d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.469ex; height:2.509ex;" alt="{\displaystyle O_{2}=2r\pi }"></span> </p><p><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 O_{1}=4d_{2}((r,0)(0,r))=4(|r-0|+|0-r|)=8r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo>−</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>8</mn> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{1}=4d_{2}((r,0)(0,r))=4(|r-0|+|0-r|)=8r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4172bfce494cbc0e39f073665cab6208f120fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.18ex; height:2.843ex;" alt="{\displaystyle O_{1}=4d_{2}((r,0)(0,r))=4(|r-0|+|0-r|)=8r}"></span> </p><p><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 O_{\infty }=4d_{\infty }((r,-r),(r,r))=max(|r-r||-r-r|)=8r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mi>r</mi> <mo>−</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>8</mn> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{\infty }=4d_{\infty }((r,-r),(r,r))=max(|r-r||-r-r|)=8r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811d87cea8f97b07543ca14f7870d9626b0404bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.647ex; height:2.843ex;" alt="{\displaystyle O_{\infty }=4d_{\infty }((r,-r),(r,r))=max(|r-r||-r-r|)=8r}"></span> </p><p>У оба <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 O_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ef134d3a8ed900181455e6b569683132371e22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{1}}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75686c667e040debea851f060bd65a4861b8cd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.649ex; height:2.509ex;" alt="{\displaystyle O_{\infty }}"></span> не појављује се <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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>. </p><p>Нека је <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77463f4fbb953a6f1fe19d83708e553f6d21457f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.066ex; height:2.343ex;" alt="{\displaystyle c_{p}}"></span> четвртина кружнице <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_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753062ca2b97967794cc23e3e553f46898493d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.484ex; height:2.843ex;" alt="{\displaystyle S_{p}}"></span> која припада првом квадранту. Тада је </p><p><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 \int _{c_{p}}ds_{p}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msub> <mi>d</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{c_{p}}ds_{p}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5667ca5b7d9e4fde08e4fedfa636b5f410264c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:9.758ex; height:6.176ex;" alt="{\displaystyle \int _{c_{p}}ds_{p}\,dx}"></span> </p><p><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 ds_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d1ba73df5e20bb3501f6ecb408adf79899f2a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.365ex; height:2.843ex;" alt="{\displaystyle ds_{p}}"></span> је елемент дужине </p><p><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 ds_{p}={\sqrt[{p}]{|dx|^{p}+|dy|^{p}}}={\sqrt[{p}]{|x'(t)|^{p}+|y'(t)|^{p}}}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>x</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>y</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds_{p}={\sqrt[{p}]{|dx|^{p}+|dy|^{p}}}={\sqrt[{p}]{|x'(t)|^{p}+|y'(t)|^{p}}}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b229ac6908f18b7681e479614393cec4d53740c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.432ex; height:4.843ex;" alt="{\displaystyle ds_{p}={\sqrt[{p}]{|dx|^{p}+|dy|^{p}}}={\sqrt[{p}]{|x'(t)|^{p}+|y'(t)|^{p}}}dt}"></span> </p><p>За <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 p=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e4100b94c1939c67f2d4b8580d26c78106c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}"></span> је </p><p><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 ds_{2}={\sqrt {(dx)^{2}+)(dy)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds_{2}={\sqrt {(dx)^{2}+)(dy)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f5ffcd36d4de5f3e8f0e7a90936c85cecfd8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:22.14ex; height:4.843ex;" alt="{\displaystyle ds_{2}={\sqrt {(dx)^{2}+)(dy)^{2}}}}"></span> </p><p>За параметризацију криве <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77463f4fbb953a6f1fe19d83708e553f6d21457f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.066ex; height:2.343ex;" alt="{\displaystyle c_{p}}"></span> узима се </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=r{\sqrt[{p}]{t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=r{\sqrt[{p}]{t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05b0a85707047cbb7551fb1e71d4a91b343dbf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.901ex; height:3.009ex;" alt="{\displaystyle x(t)=r{\sqrt[{p}]{t}}}"></span> </p><p><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 y(t)=r{\sqrt[{p}]{1-t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)=r{\sqrt[{p}]{1-t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a02314bbbe6d06306c76e783f55e39bfcba04bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.73ex; height:3.009ex;" alt="{\displaystyle y(t)=r{\sqrt[{p}]{1-t}}}"></span> za <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a5c18739ff04858eecc8fec2f53912c348e0e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:2.843ex;" alt="{\displaystyle t\in [0,1]}"></span> </p><p><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 ds_{p}={\frac {r}{p}}{\sqrt[{p}]{t^{1-p}+(1-t)^{1-p}}}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>p</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds_{p}={\frac {r}{p}}{\sqrt[{p}]{t^{1-p}+(1-t)^{1-p}}}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecca955767934c6a8f4c918e6c60579c76683afc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.5ex; height:5.509ex;" alt="{\displaystyle ds_{p}={\frac {r}{p}}{\sqrt[{p}]{t^{1-p}+(1-t)^{1-p}}}dt}"></span> </p><p>За <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 p\neq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≠</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\neq \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e8a56e74706550119d79a997e495328828cc65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.681ex; height:2.676ex;" alt="{\displaystyle p\neq \infty }"></span> </p><p><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 O_{p}={\frac {4r}{p}}\int _{0}^{1}{\sqrt[{p}]{t^{1-p}+(1-t)^{1-p}}}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>r</mi> </mrow> <mi>p</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{p}={\frac {4r}{p}}\int _{0}^{1}{\sqrt[{p}]{t^{1-p}+(1-t)^{1-p}}}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2856e1d9593e5c736b62c3eeca656ee471398561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.688ex; height:6.176ex;" alt="{\displaystyle O_{p}={\frac {4r}{p}}\int _{0}^{1}{\sqrt[{p}]{t^{1-p}+(1-t)^{1-p}}}\,dt}"></span> </p><p>Пошто су износи за <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 O_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ef134d3a8ed900181455e6b569683132371e22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{1}}"></span> i <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 O_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa32f5095ddebd2d52739d978bf85f274f900c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{2}}"></span> познати, горња формула се може проверити утврђивањем <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 p=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29a2f2fb3f642618036ed7a79712202e7ada924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e4100b94c1939c67f2d4b8580d26c78106c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}"></span>. </p><p>За <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 p=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29a2f2fb3f642618036ed7a79712202e7ada924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}"></span> </p><p><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 O_{1}=4r\int _{0}^{1}({t^{1-1}+(1-t)^{1-1})}\,dt=4r\int _{0}^{1}2\,dt=8r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>r</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>4</mn> <mi>r</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mn>2</mn> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>8</mn> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{1}=4r\int _{0}^{1}({t^{1-1}+(1-t)^{1-1})}\,dt=4r\int _{0}^{1}2\,dt=8r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb50c66b67bdc4ebbabb51dc8c92584aee205aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.453ex; height:6.176ex;" alt="{\displaystyle O_{1}=4r\int _{0}^{1}({t^{1-1}+(1-t)^{1-1})}\,dt=4r\int _{0}^{1}2\,dt=8r}"></span> </p><p>За <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 p=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e4100b94c1939c67f2d4b8580d26c78106c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}"></span> </p><p><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 O_{2}={\frac {4r}{2}}\int _{0}^{1}{\sqrt[{2}]{t^{1-2}+(1-t)^{1-2}}}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>r</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{2}={\frac {4r}{2}}\int _{0}^{1}{\sqrt[{2}]{t^{1-2}+(1-t)^{1-2}}}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb97beac7d172f33a674b4d441a7c266c820134" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.673ex; height:6.176ex;" alt="{\displaystyle O_{2}={\frac {4r}{2}}\int _{0}^{1}{\sqrt[{2}]{t^{1-2}+(1-t)^{1-2}}}\,dt}"></span> </p><p><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 O_{2}=2r\int _{0}^{1}{\sqrt[{2}]{{\frac {1}{t}}-{\frac {1}{1-t}}}}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>r</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{2}=2r\int _{0}^{1}{\sqrt[{2}]{{\frac {1}{t}}-{\frac {1}{1-t}}}}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e964c4dab5b11257af44cb41598dd5b9753f56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.714ex; height:6.343ex;" alt="{\displaystyle O_{2}=2r\int _{0}^{1}{\sqrt[{2}]{{\frac {1}{t}}-{\frac {1}{1-t}}}}\,dt}"></span> </p><p><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 O_{2}=2r\int _{0}^{1}{\sqrt[{2}]{\frac {1}{({\frac {1}{2}})^{2}-(t-{\frac {1}{2}})^{2}}}}\,dt=2r\arcsin(2t-1)|_{0}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>r</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mi>r</mi> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{2}=2r\int _{0}^{1}{\sqrt[{2}]{\frac {1}{({\frac {1}{2}})^{2}-(t-{\frac {1}{2}})^{2}}}}\,dt=2r\arcsin(2t-1)|_{0}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df365ae3170ed258ff3b81a147efa72b11230697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:54.77ex; height:7.676ex;" alt="{\displaystyle O_{2}=2r\int _{0}^{1}{\sqrt[{2}]{\frac {1}{({\frac {1}{2}})^{2}-(t-{\frac {1}{2}})^{2}}}}\,dt=2r\arcsin(2t-1)|_{0}^{1}}"></span> </p><p><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 O_{2}=2r\arcsin(1)-2r\arcsin(1)=2r\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>r</mi> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>r</mi> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>r</mi> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{2}=2r\arcsin(1)-2r\arcsin(1)=2r\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb411dd7c833fcb8f31a803ad79ef54ef528e400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.472ex; height:2.843ex;" alt="{\displaystyle O_{2}=2r\arcsin(1)-2r\arcsin(1)=2r\pi }"></span> </p><p>За сваки <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> размера <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 {O_{p}}{2r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {O_{p}}{2r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571c1b3d20f7dc8ba6077b6a992ab42e584511c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.669ex; height:5.676ex;" alt="{\displaystyle {\frac {O_{p}}{2r}}}"></span> обима <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 O_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7898d8340db13ac558a1d8e14f07f3a3b251bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.832ex; height:2.843ex;" alt="{\displaystyle O_{p}}"></span> и пречника <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 2r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb337de6fbd1ab48176084f9c4534b8c55847042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.211ex; height:2.176ex;" alt="{\displaystyle 2r}"></span> кружнице је константна. Та размера се означава са <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 \pi _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37668045f8b8d371f2e8f0d98cba26c9e7fac7d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:2.343ex;" alt="{\displaystyle \pi _{p}}"></span> и износи </p><p><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 \pi _{p}={\frac {2}{p}}\int _{0}^{1}{\sqrt[{p}]{t^{1-p}+(1-t)^{1-p}}}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{p}={\frac {2}{p}}\int _{0}^{1}{\sqrt[{p}]{t^{1-p}+(1-t)^{1-p}}}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9609e396397a70b38c8bcda6169160bfe58165a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.198ex; height:6.176ex;" alt="{\displaystyle \pi _{p}={\frac {2}{p}}\int _{0}^{1}{\sqrt[{p}]{t^{1-p}+(1-t)^{1-p}}}\,dt}"></span> </p><p>Очито је </p><p><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 \pi _{1}=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb479f88af2d0f9a6ec63279465234a5312d045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.64ex; height:2.509ex;" alt="{\displaystyle \pi _{1}=4}"></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 \pi _{2}=\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{2}=\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad40dfa725c6d5bcd101bde0d8bef88b0c6f2ec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.009ex;" alt="{\displaystyle \pi _{2}=\pi }"></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 \pi _{\infty }=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{\infty }=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b1c9ef8cbb86c298d14f6cd12ff978c1c1184c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.461ex; height:2.509ex;" alt="{\displaystyle \pi _{\infty }=4}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Референце"><span id=".D0.A0.D0.B5.D1.84.D0.B5.D1.80.D0.B5.D0.BD.D1.86.D0.B5"></span>Референце</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=13" title="Уредите одељак „Референце”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=13" title="Уреди извор одељка: Референце"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r28440201">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite id="CITEREFKatz1998" class="citation">Katz, Victor J. (1998), <i>A History of Mathematics / An Introduction</i> (2nd изд.), Addison Wesley Longman, стр. 108, <a href="/wiki/Me%C4%91unarodni_standardni_broj_knjige" title="Međunarodni standardni broj knjige">ISBN</a> <a href="/wiki/%D0%9F%D0%BE%D1%81%D0%B5%D0%B1%D0%BD%D0%BE:%D0%A8%D1%82%D0%B0%D0%BC%D0%BF%D0%B0%D0%BD%D0%B8_%D0%B8%D0%B7%D0%B2%D0%BE%D1%80%D0%B8/978-0321016188" title="Посебно:Штампани извори/978-0321016188">978-0321016188</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fsr.wikipedia.org%3A%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&rft.aufirst=Victor+J.&rft.aulast=Katz&rft.btitle=A+History+of+Mathematics+%2F+An+Introduction&rft.date=1998&rft.edition=2nd&rft.genre=book&rft.isbn=978-0321016188&rft.pages=108&rft.pub=Addison+Wesley+Longman&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation journal">Harkness, James (1898). <a rel="nofollow" class="external text" href="https://books.google.com/books/about/Introduction_to_the_Theory_of_Analytic_F.html?id=V-fVlZCc6GgC">„Introduction to the theory of analytic functions”</a>. <i>Nature</i>. <b>59</b> (1530): 30. <a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1899Natur..59..386B">1899Natur..59..386B</a>. <a href="/wiki/Digitalni_identifikator_objekta" title="Digitalni identifikator objekta">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F059386a0">10.1038/059386a0</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fsr.wikipedia.org%3A%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&rft.atitle=Introduction+to+the+theory+of+analytic+functions&rft.aufirst=James&rft.aulast=Harkness&rft.date=1898&rft.genre=article&rft.issue=1530&rft.jtitle=Nature&rft.pages=30&rft.volume=59&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%2Fabout%2FIntroduction_to_the_Theory_of_Analytic_F.html%3Fid%3DV-fVlZCc6GgC&rft_id=info%3Abibcode%2F1899Natur..59..386B&rft_id=info%3Adoi%2F10.1038%2F059386a0&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="/w/index.php?title=C._Stanley_Ogilvy&action=edit&redlink=1" class="new" title="C. Stanley Ogilvy (страница не постоји)">Ogilvy, C. Stanley</a>, <i>Excursions in Geometry</i>, Dover, 1969, 14–17.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Литература"><span id=".D0.9B.D0.B8.D1.82.D0.B5.D1.80.D0.B0.D1.82.D1.83.D1.80.D0.B0"></span>Литература</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=14" title="Уредите одељак „Литература”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=14" title="Уреди извор одељка: Литература"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r28440192">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><cite id="CITEREFPedoe1988" class="citation book">Pedoe, Dan (1988). <a rel="nofollow" class="external text" href="https://archive.org/details/geometrycomprehe0000pedo"><i>Geometry: a comprehensive course</i></a>. Dover.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fsr.wikipedia.org%3A%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&rft.aufirst=Dan&rft.aulast=Pedoe&rft.btitle=Geometry%3A+a+comprehensive+course&rft.date=1988&rft.genre=book&rft.pub=Dover&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometrycomprehe0000pedo&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/Curves/Circle.html">"Circle" in The MacTutor History of Mathematics archive</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Спољашње_везе"><span id=".D0.A1.D0.BF.D0.BE.D1.99.D0.B0.D1.88.D1.9A.D0.B5_.D0.B2.D0.B5.D0.B7.D0.B5"></span>Спољашње везе</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&veaction=edit&section=15" title="Уредите одељак „Спољашње везе”" class="mw-editsection-visualeditor"><span>уреди</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&action=edit&section=15" title="Уреди извор одељка: Спољашње везе"><span>уреди извор</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r25554621">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r25554968">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist"><span style="font-weight:bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Circle_geometry" class="extiw" title="commons:Category:Circle geometry">Кружница</a></span> на <a href="https://commons.wikimedia.org/wiki/%D0%93%D0%BB%D0%B0%D0%B2%D0%BD%D0%B0_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B0" class="extiw" title="commons:Главна страна">Викимедијиној остави</a>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25554621"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25554968"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%B7%D0%B2%D0%BE%D1%80%D0%BD%D0%B8%D0%BA" title="Викизворник">Викизворник</a> има текст из <a href="/wiki/%D0%95%D0%BD%D1%86%D0%B8%D0%BA%D0%BB%D0%BE%D0%BF%D0%B5%D0%B4%D0%B8%D1%98%D0%B0_%D0%91%D1%80%D0%B8%D1%82%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Енциклопедија Британика"><i>Енциклопедије Британике</i> (1911)</a>, чланак <i><b><a href="https://en.wikipedia.org/wiki/Wikisource:1911_Encyclop%C3%A6dia_Britannica/Circle" class="extiw" title="en:Wikisource:1911 Encyclopædia Britannica/Circle">Circle</a></b></i>.</div></div> </div> <ul><li><cite class="citation book">Hazewinkel Michiel, ур. (2001). „Circle”. <a rel="nofollow" class="external text" href="http://eom.springer.de/p/c022260.htm"><i>Encyclopaedia of Mathematics</i></a>. Springer. <a href="/wiki/Me%C4%91unarodni_standardni_broj_knjige" title="Međunarodni standardni broj knjige">ISBN</a> <a href="/wiki/%D0%9F%D0%BE%D1%81%D0%B5%D0%B1%D0%BD%D0%BE:%D0%A8%D1%82%D0%B0%D0%BC%D0%BF%D0%B0%D0%BD%D0%B8_%D0%B8%D0%B7%D0%B2%D0%BE%D1%80%D0%B8/978-1556080104" title="Посебно:Штампани извори/978-1556080104">978-1556080104</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fsr.wikipedia.org%3A%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&rft.atitle=Circle&rft.btitle=Encyclopaedia+of+Mathematics&rft.date=2001&rft.genre=bookitem&rft.pub=Springer&rft_id=http%3A%2F%2Feom.springer.de%2Fp%2Fc022260.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a href="https://planetmath.org/alphabetical.html" class="extiw" title="planetmath:4236">Circle (PlanetMath.org website)</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-Circle"><cite class="citation web">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Circle.html">„Circle”</a>. <i><a href="/w/index.php?title=MathWorld&action=edit&redlink=1" class="new" title="MathWorld (страница не постоји)">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fsr.wikipedia.org%3A%D0%9A%D1%80%D1%83%D0%B6%D0%BD%D0%B8%D1%86%D0%B0&rft.atitle=Circle&rft.au=Weisstein%2C+Eric+W.&rft.genre=unknown&rft.jtitle=MathWorld&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FCircle.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/tocs/circlestoc.html">Interactive Java applets</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathwarehouse.com/geometry/circle/interactive-circle-equation.php">Interactive Standard Form Equation of Circle</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/pythagoras/Munching/circle.shtml">Munching on Circles</a> at <a href="/w/index.php?title=Cut-the-knot&action=edit&redlink=1" class="new" title="Cut-the-knot (страница не постоји)">cut-the-knot</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r25469611">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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role="navigation" class="navbox authority-control" aria-labelledby="Нормативна_контрола_text-top&#124;10px&#124;alt=Уреди_на_Википодацима&#124;link=https&#58;//www.wikidata.org/wiki/Q17278#identifiers&#124;class=noprint&#124;Уреди_на_Википодацима" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Нормативна_контрола_text-top&#124;10px&#124;alt=Уреди_на_Википодацима&#124;link=https&#58;//www.wikidata.org/wiki/Q17278#identifiers&#124;class=noprint&#124;Уреди_на_Википодацима" style="font-size:114%;margin:0 4em"><a href="/wiki/%D0%9F%D0%BE%D0%BC%D0%BE%D1%9B:%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%B0_%D0%BA%D0%BE%D0%BD%D1%82%D1%80%D0%BE%D0%BB%D0%B0" title="Помоћ:Нормативна контрола">Нормативна контрола</a> <span class="mw-valign-text-top noprint" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q17278#identifiers" title="Уреди на Википодацима"><img alt="Уреди на Википодацима" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Државне</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4032962-8">Немачка</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="http://uli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007286430105171">Израел</a></span></li> <li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Circle"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85026057">Сједињене Државе</a></span></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000321923&P_CON_LNG=ENG">Летонија</a></span></li> <li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="kružnice"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph121971&CON_LNG=ENG">Чешка</a></span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Остале</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="circle"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/circle-mathematics">Енциклопедија Британика</a></span></span></li></ul> </div></td></tr></tbody></table></div>    </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="">Преузето из „<a dir="ltr" href="https://sr.wikipedia.org/w/index.php?title=Кружница&oldid=27550269">https://sr.wikipedia.org/w/index.php?title=Кружница&oldid=27550269</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%9F%D0%BE%D1%81%D0%B5%D0%B1%D0%BD%D0%BE:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B5" title="Посебно:Категорије">Категорије</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%9A%D1%80%D1%83%D0%B3%D0%BE%D0%B2%D0%B8" title="Категорија:Кругови">Кругови</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%9F%D0%B8" title="Категорија:Пи">Пи</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%9A%D0%BE%D0%BD%D1%83%D1%81%D0%BD%D0%B8_%D0%BF%D1%80%D0%B5%D1%81%D0%B5%D1%86%D0%B8" title="Категорија:Конусни пресеци">Конусни пресеци</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%9A%D1%80%D0%B8%D0%B2%D0%B5" title="Категорија:Криве">Криве</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%95%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D0%B8_%D0%BE%D0%B1%D0%BB%D0%B8%D1%86%D0%B8" title="Категорија:Елементарни облици">Елементарни облици</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Сакривене категорије: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%A7%D0%BB%D0%B0%D0%BD%D1%86%D0%B8_%D0%BA%D0%BE%D1%98%D0%B8_%D0%BA%D0%BE%D1%80%D0%B8%D1%81%D1%82%D0%B5_%D1%88%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD%D0%B5_%D0%B7%D0%B0_%D0%B8%D0%BD%D1%84%D0%BE%D0%BA%D1%83%D1%82%D0%B8%D1%98%D0%B5_%D0%B1%D0%B5%D0%B7_%D1%80%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0_%D0%BF%D0%BE%D0%B4%D0%B0%D1%82%D0%B0%D0%BA%D0%B0" title="Категорија:Чланци који користе шаблоне за инфокутије без редова података">Чланци који користе шаблоне за инфокутије без редова података</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%A8%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BD%D0%B0_%D0%9E%D1%81%D1%82%D0%B0%D0%B2%D0%B8/%D0%BF%D0%B0%D1%80%D0%B0%D0%BC%D0%B5%D1%82%D0%B0%D1%80/%D0%BD%D0%B5%D0%BD%D0%B0%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD/%D0%B8%D0%BC%D0%B5_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B5_%D1%80%D0%B0%D0%B7%D0%BB%D0%B8%D1%87%D0%B8%D1%82%D0%BE_%D0%BE%D0%B4_%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%BE%D0%B4%D0%B0%D1%82%D0%B0%D0%BA%D0%B0" title="Категорија:Шаблон:Категорија на Остави/параметар/ненаведен/име странице различито од Википодатака">Шаблон:Категорија на Остави/параметар/ненаведен/име странице различито од Википодатака</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%A8%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BD%D0%B0_%D0%9E%D1%81%D1%82%D0%B0%D0%B2%D0%B8/%D0%B8%D0%BC%D0%B5%D0%BD%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80/%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D0%B8" title="Категорија:Шаблон:Категорија на Остави/именски простор/главни">Шаблон:Категорија на Остави/именски простор/главни</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%A7%D0%BB%D0%B0%D0%BD%D1%86%D0%B8_%D1%81%D0%B0_GND_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BC%D0%B0" title="Категорија:Чланци са GND идентификаторима">Чланци са GND идентификаторима</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%A7%D0%BB%D0%B0%D0%BD%D1%86%D0%B8_%D1%81%D0%B0_J9U_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BC%D0%B0" title="Категорија:Чланци са J9U идентификаторима">Чланци са J9U идентификаторима</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%A7%D0%BB%D0%B0%D0%BD%D1%86%D0%B8_%D1%81%D0%B0_LCCN_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BC%D0%B0" title="Категорија:Чланци са LCCN идентификаторима">Чланци са LCCN идентификаторима</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%A7%D0%BB%D0%B0%D0%BD%D1%86%D0%B8_%D1%81%D0%B0_LNB_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BC%D0%B0" title="Категорија:Чланци са LNB идентификаторима">Чланци са LNB идентификаторима</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%A7%D0%BB%D0%B0%D0%BD%D1%86%D0%B8_%D1%81%D0%B0_NKC_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BC%D0%B0" title="Категорија:Чланци са NKC идентификаторима">Чланци са NKC идентификаторима</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0:%D0%A7%D0%BB%D0%B0%D0%BD%D1%86%D0%B8_%D1%81%D0%B0_EBO_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BC%D0%B0" title="Категорија:Чланци са EBO идентификаторима">Чланци са EBO идентификаторима</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"> Датум и време последње измене странице: 7. април 2024. у 16:00.</li> <li id="footer-info-copyright">Текст је доступан под лиценцом <a rel="nofollow" class="external text" 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