CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Kite (geometry) - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"0e9a12cd-51c3-471f-8095-d50c68340055","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Kite_(geometry)","wgTitle":"Kite (geometry)","wgCurRevisionId":1257445490,"wgRevisionId":1257445490,"wgArticleId":84840,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 maint: location missing publisher","Articles with short description","Short description is different from Wikidata","Good articles","Use dmy dates from October 2022","Use list-defined references from October 2022","Pages using multiple image with auto scaled images","Commons category link is on Wikidata","Elementary shapes","Types of quadrilaterals","Kites"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName": "Kite_(geometry)","wgRelevantArticleId":84840,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgRedirectedFrom":"Dart_(geometry)","wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":40000,"wgInternalRedirectTargetUrl":"/wiki/Kite_(geometry)","wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q107061", "wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site", "mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/80/GeometricKite.svg/1200px-GeometricKite.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1394"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/80/GeometricKite.svg/800px-GeometricKite.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="929"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/80/GeometricKite.svg/640px-GeometricKite.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="743"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Kite (geometry) - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Kite_(geometry)"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Kite_(geometry)&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Kite_(geometry)"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Kite_geometry rootpage-Kite_geometry skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Kite+%28geometry%29" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Kite+%28geometry%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Kite+%28geometry%29" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Kite+%28geometry%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <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="Contents" 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">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</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">(Top)</div> </a> </li> <li id="toc-Definition_and_classification" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_and_classification"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition and classification</span> </div> </a> <ul id="toc-Definition_and_classification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_cases" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Special_cases"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Special cases</span> </div> </a> <ul id="toc-Special_cases-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Diagonals,_angles,_and_area" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diagonals,_angles,_and_area"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Diagonals, angles, and area</span> </div> </a> <ul id="toc-Diagonals,_angles,_and_area-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inscribed_circle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inscribed_circle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Inscribed circle</span> </div> </a> <ul id="toc-Inscribed_circle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Duality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Duality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Duality</span> </div> </a> <ul id="toc-Duality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dissection" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dissection"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Dissection</span> </div> </a> <ul id="toc-Dissection-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tilings_and_polyhedra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tilings_and_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Tilings and polyhedra</span> </div> </a> <ul id="toc-Tilings_and_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Outer_billiards" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Outer_billiards"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Outer billiards</span> </div> </a> <ul id="toc-Outer_billiards-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Kite (geometry)</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="Go to an article in another language. Available in 59 languages" > <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-59" 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">59 languages</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/Vlie%C3%ABr_(meetkunde)" title="Vlieër (meetkunde) – Afrikaans" lang="af" hreflang="af" data-title="Vlieër (meetkunde)" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%A7%D8%A6%D8%B1%D8%A9_%D9%88%D8%B1%D9%82%D9%8A%D8%A9_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9)" title="طائرة ورقية (هندسة رياضية) – Arabic" lang="ar" hreflang="ar" data-title="طائرة ورقية (هندسة رياضية)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Deltoide" title="Deltoide – Asturian" lang="ast" hreflang="ast" data-title="Deltoide" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Deltoid" title="Deltoid – Azerbaijani" lang="az" hreflang="az" data-title="Deltoid" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8D%D0%BB%D1%8C%D1%82%D0%BE%D1%96%D0%B4" title="Дэльтоід – Belarusian" lang="be" hreflang="be" data-title="Дэльтоід" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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%94%D1%8D%D0%BB%D1%8C%D1%82%D0%BE%D1%96%D0%B4" title="Дэльтоід – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Дэльтоід" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%82%D0%BE%D0%B8%D0%B4" title="Делтоид – Bulgarian" lang="bg" hreflang="bg" data-title="Делтоид" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Deltoid" title="Deltoid – Bosnian" lang="bs" hreflang="bs" data-title="Deltoid" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Deltoide" title="Deltoide – Catalan" lang="ca" hreflang="ca" data-title="Deltoide" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Deltoid" title="Deltoid – Czech" lang="cs" hreflang="cs" data-title="Deltoid" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Kayiti" title="Kayiti – Shona" lang="sn" hreflang="sn" data-title="Kayiti" data-language-autonym="ChiShona" data-language-local-name="Shona" 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/Barcut_(geometreg)" title="Barcut (geometreg) – Welsh" lang="cy" hreflang="cy" data-title="Barcut (geometreg)" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Dragefirkant" title="Dragefirkant – Danish" lang="da" hreflang="da" data-title="Dragefirkant" data-language-autonym="Dansk" data-language-local-name="Danish" 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/Drachenviereck" title="Drachenviereck – German" lang="de" hreflang="de" data-title="Drachenviereck" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Romboid" title="Romboid – Estonian" lang="et" hreflang="et" data-title="Romboid" data-language-autonym="Eesti" data-language-local-name="Estonian" 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%94%CE%B5%CE%BB%CF%84%CE%BF%CE%B5%CE%B9%CE%B4%CE%AD%CF%82" title="Δελτοειδές – Greek" lang="el" hreflang="el" data-title="Δελτοειδές" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Deltoide" title="Deltoide – Spanish" lang="es" hreflang="es" data-title="Deltoide" data-language-autonym="Español" data-language-local-name="Spanish" 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/Kajto_(geometrio)" title="Kajto (geometrio) – Esperanto" lang="eo" hreflang="eo" data-title="Kajto (geometrio)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" 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/Kometa_(geometria)" title="Kometa (geometria) – Basque" lang="eu" hreflang="eu" data-title="Kometa (geometria)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D8%A7%DB%8C%D8%AA_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="کایت (هندسه) – Persian" lang="fa" hreflang="fa" data-title="کایت (هندسه)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cerf-volant_(g%C3%A9om%C3%A9trie)" title="Cerf-volant (géométrie) – French" lang="fr" hreflang="fr" data-title="Cerf-volant (géométrie)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%B0%EA%BC%B4" title="연꼴 – Korean" lang="ko" hreflang="ko" data-title="연꼴" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Layang-layang_(geometri)" title="Layang-layang (geometri) – Indonesian" lang="id" hreflang="id" data-title="Layang-layang (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Deltoide" title="Deltoide – Interlingua" lang="ia" hreflang="ia" data-title="Deltoide" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Aquilone_(geometria)" title="Aquilone (geometria) – Italian" lang="it" hreflang="it" data-title="Aquilone (geometria)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%9C%D7%AA%D7%95%D7%9F" title="דלתון – Hebrew" lang="he" hreflang="he" data-title="דלתון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Layang-layang_(g%C3%A9om%C3%A8tri)" title="Layang-layang (géomètri) – Javanese" lang="jv" hreflang="jv" data-title="Layang-layang (géomètri)" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%BE%D0%B8%D0%B4" title="Дельтоид – Kazakh" lang="kk" hreflang="kk" data-title="Дельтоид" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Delto%C4%ABds" title="Deltoīds – Latvian" lang="lv" hreflang="lv" data-title="Deltoīds" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Deltoid" title="Deltoid – Hungarian" lang="hu" hreflang="hu" data-title="Deltoid" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%82%D0%BE%D0%B8%D0%B4" title="Делтоид – Macedonian" lang="mk" hreflang="mk" data-title="Делтоид" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Layang-layang_(geometri)" title="Layang-layang (geometri) – Malay" lang="ms" hreflang="ms" data-title="Layang-layang (geometri)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vlieger_(meetkunde)" title="Vlieger (meetkunde) – Dutch" lang="nl" hreflang="nl" data-title="Vlieger (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%87%A7%E5%BD%A2" title="凧形 – Japanese" lang="ja" hreflang="ja" data-title="凧形" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Drage_(geometri)" title="Drage (geometri) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Drage (geometri)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" 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/Drake_i_geometrien" title="Drake i geometrien – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Drake i geometrien" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A8%A4%E0%A9%B0%E0%A8%97_(%E0%A8%B0%E0%A9%87%E0%A8%96%E0%A8%BE_%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4)" title="ਪਤੰਗ (ਰੇਖਾ ਗਣਿਤ) – Punjabi" lang="pa" hreflang="pa" data-title="ਪਤੰਗ (ਰੇਖਾ ਗਣਿਤ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" 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/Draken_(Geometrie)" title="Draken (Geometrie) – Low German" lang="nds" hreflang="nds" data-title="Draken (Geometrie)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Deltoid" title="Deltoid – Polish" lang="pl" hreflang="pl" data-title="Deltoid" data-language-autonym="Polski" data-language-local-name="Polish" 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/Deltoide" title="Deltoide – Portuguese" lang="pt" hreflang="pt" data-title="Deltoide" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Romboid" title="Romboid – Romanian" lang="ro" hreflang="ro" data-title="Romboid" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%BE%D0%B8%D0%B4" title="Дельтоид – Russian" lang="ru" hreflang="ru" data-title="Дельтоид" data-language-autonym="Русский" data-language-local-name="Russian" 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/Kite_(geometry)" title="Kite (geometry) – Scots" lang="sco" hreflang="sco" data-title="Kite (geometry)" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Balona_(gjeometri)" title="Balona (gjeometri) – Albanian" lang="sq" hreflang="sq" data-title="Balona (gjeometri)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Deltoid" title="Deltoid – Slovak" lang="sk" hreflang="sk" data-title="Deltoid" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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/Deltoid" title="Deltoid – Slovenian" lang="sl" hreflang="sl" data-title="Deltoid" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%DB%86%D9%84%D8%A7%D8%B1%DB%95_(%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95)" title="کۆلارە (ئەندازە) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="کۆلارە (ئەندازە)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%82%D0%BE%D0%B8%D0%B4" title="Делтоид – Serbian" lang="sr" hreflang="sr" data-title="Делтоид" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Deltoid" title="Deltoid – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Deltoid" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" 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/Langlayangan_(%C3%A9lmu_ukur)" title="Langlayangan (élmu ukur) – Sundanese" lang="su" hreflang="su" data-title="Langlayangan (élmu ukur)" data-language-autonym="Sunda" data-language-local-name="Sundanese" 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/Leija_(geometria)" title="Leija (geometria) – Finnish" lang="fi" hreflang="fi" data-title="Leija (geometria)" data-language-autonym="Suomi" data-language-local-name="Finnish" 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/Drake_(geometri)" title="Drake (geometri) – Swedish" lang="sv" hreflang="sv" data-title="Drake (geometri)" data-language-autonym="Svenska" data-language-local-name="Swedish" 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%AA%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D_(%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="பட்டம் (வடிவவியல்) – Tamil" lang="ta" hreflang="ta" data-title="பட்டம் (வடிவவியல்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</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%AA%E0%B8%B5%E0%B9%88%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%A7%E0%B9%88%E0%B8%B2%E0%B8%A7" title="รูปสี่เหลี่ยมรูปว่าว – Thai" lang="th" hreflang="th" data-title="รูปสี่เหลี่ยมรูปว่าว" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Deltoid_(geometri)" title="Deltoid (geometri) – Turkish" lang="tr" hreflang="tr" data-title="Deltoid (geometri)" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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%94%D0%B5%D0%BB%D1%8C%D1%82%D0%BE%D1%97%D0%B4" title="Дельтоїд – Ukrainian" lang="uk" hreflang="uk" data-title="Дельтоїд" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%AD%9D%E5%BD%A2" title="筝形 – Wu" lang="wuu" hreflang="wuu" data-title="筝形" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%B7%82%E5%BD%A2" title="鷂形 – Cantonese" lang="yue" hreflang="yue" data-title="鷂形" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%B7%82%E5%BD%A2" title="鷂形 – Chinese" lang="zh" hreflang="zh" data-title="鷂形" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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/Q107061#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Kite_(geometry)" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Kite_(geometry)" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Kite_(geometry)"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Kite_(geometry)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Kite_(geometry)&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Kite_(geometry)"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Kite_(geometry)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Kite_(geometry)&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Kite_(geometry)" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Kite_(geometry)" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Kite_(geometry)&oldid=1257445490" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Kite_(geometry)&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Kite_%28geometry%29&id=1257445490&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FKite_%28geometry%29"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FKite_%28geometry%29"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Kite_%28geometry%29&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Kite_(geometry)&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</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"> In other projects </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:Deltoids" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/A_Guide_to_the_GRE/Kites" hreflang="en"><span>Wikibooks</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/Q107061" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</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="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <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">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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 id="mw-indicator-good-star" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Wikipedia:Good_articles*" title="This is a good article. Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Dart_(geometry)&redirect=no" class="mw-redirect" title="Dart (geometry)">Dart (geometry)</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Quadrilateral symmetric across a diagonal</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.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%}@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;">Kite</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:GeometricKite.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/GeometricKite.svg/220px-GeometricKite.svg.png" decoding="async" width="220" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/GeometricKite.svg/330px-GeometricKite.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/GeometricKite.svg/440px-GeometricKite.svg.png 2x" data-file-width="310" data-file-height="360" /></a></span><div class="infobox-caption">A kite, showing its pairs of equal-length sides and its inscribed circle.</div></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a> and <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a></th><td class="infobox-data">4</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_planar_symmetry_groups" title="List of planar symmetry groups">Symmetry group</a></th><td class="infobox-data"><a href="/wiki/Reflection_symmetry" title="Reflection symmetry"><i>D</i><sub><i>1</i></sub></a> (*)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dual_polygon" title="Dual polygon">Dual polygon</a></th><td class="infobox-data"><a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">Isosceles trapezoid</a></td></tr></tbody></table> <p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, a <b>kite </b>is a <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> with <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a> across a <a href="/wiki/Diagonal" title="Diagonal">diagonal</a>. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as <b>deltoids</b>,<sup id="cite_ref-halsted_1-0" class="reference"><a href="#cite_note-halsted-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> but the word <i>deltoid</i> may also refer to a <a href="/wiki/Deltoid_curve" title="Deltoid curve">deltoid curve</a>, an unrelated geometric object sometimes studied in connection with quadrilaterals.<sup id="cite_ref-goormaghtigh_2-0" class="reference"><a href="#cite_note-goormaghtigh-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> A kite may also be called a <b>dart</b>,<sup id="cite_ref-charter-rogers_4-0" class="reference"><a href="#cite_note-charter-rogers-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> particularly if it is not convex.<sup id="cite_ref-gardner_5-0" class="reference"><a href="#cite_note-gardner-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-thurston_6-0" class="reference"><a href="#cite_note-thurston-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Every kite is an <a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">orthodiagonal quadrilateral</a> (its diagonals are at right angles) and, when convex, a <a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">tangential quadrilateral</a> (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and tangential. They include as special cases the <a href="/wiki/Right_kite" title="Right kite">right kites</a>, with two opposite right angles; the <a href="/wiki/Rhombus" title="Rhombus">rhombi</a>, with two diagonal axes of symmetry; and the <a href="/wiki/Square" title="Square">squares</a>, which are also special cases of both right kites and rhombi. </p><p>The quadrilateral with the greatest ratio of <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> to <a href="/wiki/Diameter" title="Diameter">diameter</a> is a kite, with 60°, 75°, and 150° angles. Kites of two shapes (one convex and one non-convex) form the <a href="/wiki/Prototile" title="Prototile">prototiles</a> of one of the forms of the <a href="/wiki/Penrose_tiling" title="Penrose tiling">Penrose tiling</a>. Kites also form the faces of several <a href="/wiki/Isohedral_figure" title="Isohedral figure">face-symmetric</a> polyhedra and <a href="/wiki/Tessellation" title="Tessellation">tessellations</a>, and have been studied in connection with <a href="/wiki/Outer_billiards" title="Outer billiards">outer billiards</a>, a problem in the advanced mathematics of <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_classification">Definition and classification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=1" title="Edit section: Definition and classification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Deltoid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Deltoid.svg/220px-Deltoid.svg.png" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Deltoid.svg/330px-Deltoid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Deltoid.svg/440px-Deltoid.svg.png 2x" data-file-width="839" data-file-height="514" /></a><figcaption>Convex and concave kites</figcaption></figure> <p>A kite is a <a href="/wiki/Quadrilateral" title="Quadrilateral">quadrilateral</a> with <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a> across one of its diagonals. Equivalently, it is a quadrilateral whose four sides can be grouped into two pairs of adjacent equal-length sides.<sup id="cite_ref-halsted_1-1" class="reference"><a href="#cite_note-halsted-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-devilliers-adventures_7-0" class="reference"><a href="#cite_note-devilliers-adventures-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> A kite can be constructed from the centers and crossing points of any two intersecting <a href="/wiki/Circle" title="Circle">circles</a>.<sup id="cite_ref-idiot_8-0" class="reference"><a href="#cite_note-idiot-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Kites as described here may be either <a href="/wiki/Convex_polygon" title="Convex polygon">convex</a> or <a href="/wiki/Concave_polygon" title="Concave polygon">concave</a>, although some sources restrict <i>kite</i> to mean only convex kites. A quadrilateral is a kite <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> any one of the following conditions is true: </p> <ul><li>The four sides can be split into two pairs of adjacent equal-length sides.<sup id="cite_ref-devilliers-adventures_7-1" class="reference"><a href="#cite_note-devilliers-adventures-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>One diagonal crosses the midpoint of the other diagonal at a right angle, forming its <a href="/wiki/Perpendicular_bisector" class="mw-redirect" title="Perpendicular bisector">perpendicular bisector</a>.<sup id="cite_ref-usiskin-griffin_9-0" class="reference"><a href="#cite_note-usiskin-griffin-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> (In the concave case, the line through one of the diagonals bisects the other.)</li> <li>One diagonal is a line of symmetry. It divides the quadrilateral into two congruent triangles that are mirror images of each other.<sup id="cite_ref-devilliers-adventures_7-2" class="reference"><a href="#cite_note-devilliers-adventures-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>One diagonal <a href="/wiki/Angle_bisector" class="mw-redirect" title="Angle bisector">bisects</a> both of the angles at its two ends.<sup id="cite_ref-devilliers-adventures_7-3" class="reference"><a href="#cite_note-devilliers-adventures-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul> <p>Kite quadrilaterals are named for the wind-blown, flying <a href="/wiki/Kite" title="Kite">kites</a>, which often have this shape<sup id="cite_ref-beamer_10-0" class="reference"><a href="#cite_note-beamer-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-alexander-koeberlein_11-0" class="reference"><a href="#cite_note-alexander-koeberlein-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and which are in turn named for <a href="/wiki/Kite_(bird)" title="Kite (bird)">a hovering bird</a> and the sound it makes.<sup id="cite_ref-nuncius_12-0" class="reference"><a href="#cite_note-nuncius-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-liberman_13-0" class="reference"><a href="#cite_note-liberman-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> According to <a href="/wiki/Olaus_Henrici" title="Olaus Henrici">Olaus Henrici</a>, the name "kite" was given to these shapes by <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">James Joseph Sylvester</a>.<sup id="cite_ref-henrici_14-0" class="reference"><a href="#cite_note-henrici-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Quadrilaterals can be classified <i>hierarchically</i>, meaning that some classes of quadrilaterals include other classes, or <i>partitionally</i>, meaning that each quadrilateral is in only one class. Classified hierarchically, kites include the <a href="/wiki/Rhombus" title="Rhombus">rhombi</a> (quadrilaterals with four equal sides) and <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">squares</a>. All <a href="/wiki/Equilateral_polygon" title="Equilateral polygon">equilateral</a> kites are rhombi, and all <a href="/wiki/Equiangular_polygon" title="Equiangular polygon">equiangular</a> kites are squares. When classified partitionally, rhombi and squares would not be kites, because they belong to a different class of quadrilaterals; similarly, the <a href="/wiki/Right_kite" title="Right kite">right kites</a> discussed below would not be kites. The remainder of this article follows a hierarchical classification; rhombi, squares, and right kites are all considered kites. By avoiding the need to consider special cases, this classification can simplify some facts about kites.<sup id="cite_ref-devilliers-role_15-0" class="reference"><a href="#cite_note-devilliers-role-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Like kites, a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent. Any <a href="/wiki/Simple_polygon" title="Simple polygon">non-self-crossing</a> quadrilateral that has an axis of symmetry must be either a kite, with a diagonal axis of symmetry; or an <a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">isosceles trapezoid</a>, with an axis of symmetry through the midpoints of two sides. These include as special cases the <a href="/wiki/Rhombus" title="Rhombus">rhombus</a> and the <a href="/wiki/Rectangle" title="Rectangle">rectangle</a> respectively, and the square, which is a special case of both.<sup id="cite_ref-halsted_1-2" class="reference"><a href="#cite_note-halsted-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The self-crossing quadrilaterals include another class of symmetric quadrilaterals, the <a href="/wiki/Antiparallelogram" title="Antiparallelogram">antiparallelograms</a>.<sup id="cite_ref-alsina-nelson_16-0" class="reference"><a href="#cite_note-alsina-nelson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=2" title="Edit section: Special cases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:591px;max-width:591px"><div class="trow"><div class="tsingle" style="width:183px;max-width:183px"><div class="thumbimage" style="height:183px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Bicentric_kite_001.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Bicentric_kite_001.svg/181px-Bicentric_kite_001.svg.png" decoding="async" width="181" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Bicentric_kite_001.svg/272px-Bicentric_kite_001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Bicentric_kite_001.svg/362px-Bicentric_kite_001.svg.png 2x" data-file-width="426" data-file-height="431" /></a></span></div><div class="thumbcaption">Right kite</div></div><div class="tsingle" style="width:185px;max-width:185px"><div class="thumbimage" style="height:183px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Reuleaux_kite.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Reuleaux_kite.svg/183px-Reuleaux_kite.svg.png" decoding="async" width="183" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Reuleaux_kite.svg/275px-Reuleaux_kite.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Reuleaux_kite.svg/366px-Reuleaux_kite.svg.png 2x" data-file-width="363" data-file-height="363" /></a></span></div><div class="thumbcaption">Equidiagonal kite in a <a href="/wiki/Reuleaux_triangle" title="Reuleaux triangle">Reuleaux triangle</a></div></div><div class="tsingle" style="width:217px;max-width:217px"><div class="thumbimage" style="height:183px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Lute_of_Pythagoras.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Lute_of_Pythagoras.svg/215px-Lute_of_Pythagoras.svg.png" decoding="async" width="215" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Lute_of_Pythagoras.svg/323px-Lute_of_Pythagoras.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Lute_of_Pythagoras.svg/430px-Lute_of_Pythagoras.svg.png 2x" data-file-width="1905" data-file-height="1621" /></a></span></div><div class="thumbcaption"><a href="/wiki/Lute_of_Pythagoras" title="Lute of Pythagoras">Lute of Pythagoras</a></div></div></div></div></div> <p>The <a href="/wiki/Right_kite" title="Right kite">right kites</a> have two opposite <a href="/wiki/Right_angle" title="Right angle">right angles</a>.<sup id="cite_ref-devilliers-role_15-1" class="reference"><a href="#cite_note-devilliers-role-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-alsina-nelson_16-1" class="reference"><a href="#cite_note-alsina-nelson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The right kites are exactly the kites that are <a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilaterals</a>, meaning that there is a circle that passes through all their vertices.<sup id="cite_ref-gant_17-0" class="reference"><a href="#cite_note-gant-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The cyclic quadrilaterals may equivalently defined as the quadrilaterals in which two opposite angles are <a href="/wiki/Supplementary_angles" class="mw-redirect" title="Supplementary angles">supplementary</a> (they add to 180°); if one pair is supplementary the other is as well.<sup id="cite_ref-usiskin-griffin_9-1" class="reference"><a href="#cite_note-usiskin-griffin-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Therefore, the right kites are the kites with two opposite supplementary angles, for either of the two opposite pairs of angles. Because right kites circumscribe one circle and are inscribed in another circle, they are <a href="/wiki/Bicentric_quadrilateral" title="Bicentric quadrilateral">bicentric quadrilaterals</a> (actually tricentric, as they also have a third circle externally tangent to the <a href="/wiki/Extended_side" title="Extended side">extensions of their sides</a>).<sup id="cite_ref-alsina-nelson_16-2" class="reference"><a href="#cite_note-alsina-nelson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> If the sizes of an inscribed and a circumscribed circle are fixed, the right kite has the largest area of any quadrilateral trapped between them.<sup id="cite_ref-josefsson-area_18-0" class="reference"><a href="#cite_note-josefsson-area-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Among all quadrilaterals, the shape that has the greatest ratio of its <a href="/wiki/Perimeter" title="Perimeter">perimeter</a> to its <a href="/wiki/Diameter" title="Diameter">diameter</a> (maximum distance between any two points) is an <a href="/wiki/Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">equidiagonal</a> kite with angles 60°, 75°, 150°, 75°. Its four vertices lie at the three corners and one of the side midpoints of the <a href="/wiki/Reuleaux_triangle" title="Reuleaux triangle">Reuleaux triangle</a>.<sup id="cite_ref-ball_19-0" class="reference"><a href="#cite_note-ball-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-griffiths-culpin_20-0" class="reference"><a href="#cite_note-griffiths-culpin-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> When an equidiagonal kite has side lengths less than or equal to its diagonals, like this one or the square, it is one of the quadrilaterals with the <a href="/wiki/Biggest_little_polygon" title="Biggest little polygon">greatest ratio of area to diameter</a>.<sup id="cite_ref-audet-hansen-svrtan_21-0" class="reference"><a href="#cite_note-audet-hansen-svrtan-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>A kite with three 108° angles and one 36° angle forms the <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of the <a href="/wiki/Lute_of_Pythagoras" title="Lute of Pythagoras">lute of Pythagoras</a>, a <a href="/wiki/Fractal" title="Fractal">fractal</a> made of nested <a href="/wiki/Pentagram" title="Pentagram">pentagrams</a>.<sup id="cite_ref-darling_22-0" class="reference"><a href="#cite_note-darling-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The four sides of this kite lie on four of the sides of a <a href="/wiki/Regular_pentagon" class="mw-redirect" title="Regular pentagon">regular pentagon</a>, with a <a href="/wiki/Golden_triangle_(mathematics)" title="Golden triangle (mathematics)">golden triangle</a> glued onto the fifth side.<sup id="cite_ref-alsina-nelson_16-3" class="reference"><a href="#cite_note-alsina-nelson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Aperiodic_monotile_smith_2023.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Aperiodic_monotile_smith_2023.svg/220px-Aperiodic_monotile_smith_2023.svg.png" decoding="async" width="220" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Aperiodic_monotile_smith_2023.svg/330px-Aperiodic_monotile_smith_2023.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Aperiodic_monotile_smith_2023.svg/440px-Aperiodic_monotile_smith_2023.svg.png 2x" data-file-width="510" data-file-height="431" /></a><figcaption>Part of an aperiodic tiling with prototiles made from eight kites</figcaption></figure> <p>There are only eight polygons that can tile the plane such that reflecting any tile across any one of its edges produces another tile; this arrangement is called an <a href="/wiki/Edge_tessellation" title="Edge tessellation">edge tessellation</a>. One of them is a tiling by a right kite, with 60°, 90°, and 120° angles. It produces the <a href="/wiki/Deltoidal_trihexagonal_tiling" class="mw-redirect" title="Deltoidal trihexagonal tiling">deltoidal trihexagonal tiling</a> (see <a href="#Tilings_and_polyhedra">§ Tilings and polyhedra</a>).<sup id="cite_ref-kirby-umble_23-0" class="reference"><a href="#cite_note-kirby-umble-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> A <a href="/wiki/Prototile" title="Prototile">prototile</a> made by eight of these kites tiles the plane only <a href="/wiki/Aperiodic_tiling" title="Aperiodic tiling">aperiodically</a>, key to a claimed solution of the <a href="/wiki/Einstein_problem" title="Einstein problem">einstein problem</a>.<sup id="cite_ref-smkg_24-0" class="reference"><a href="#cite_note-smkg-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>, a kite can have three right angles and one non-right angle, forming a special case of a <a href="/wiki/Lambert_quadrilateral" title="Lambert quadrilateral">Lambert quadrilateral</a>. The fourth angle is acute in <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> and obtuse in <a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a>.<sup id="cite_ref-eves_25-0" class="reference"><a href="#cite_note-eves-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Diagonals,_angles,_and_area"><span id="Diagonals.2C_angles.2C_and_area"></span>Diagonals, angles, and area</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=4" title="Edit section: Diagonals, angles, and area"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every kite is an <a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">orthodiagonal quadrilateral</a>, meaning that its two diagonals are <a href="/wiki/Perpendicular" title="Perpendicular">at right angles</a> to each other. Moreover, one of the two diagonals (the symmetry axis) is the <a href="/wiki/Perpendicular_bisector" class="mw-redirect" title="Perpendicular bisector">perpendicular bisector</a> of the other, and is also the <a href="/wiki/Angle_bisector" class="mw-redirect" title="Angle bisector">angle bisector</a> of the two angles it meets.<sup id="cite_ref-halsted_1-3" class="reference"><a href="#cite_note-halsted-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Because of its symmetry, the other two angles of the kite must be equal.<sup id="cite_ref-beamer_10-1" class="reference"><a href="#cite_note-beamer-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-alexander-koeberlein_11-1" class="reference"><a href="#cite_note-alexander-koeberlein-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The diagonal symmetry axis of a convex kite divides it into two <a href="/wiki/Congruent_triangles" class="mw-redirect" title="Congruent triangles">congruent triangles</a>; the other diagonal divides it into two <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles triangles</a>.<sup id="cite_ref-halsted_1-4" class="reference"><a href="#cite_note-halsted-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>As is true more generally for any orthodiagonal quadrilateral, the area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> of a kite may be calculated as half the product of the lengths of the diagonals <span class="mwe-math-element"><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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>:<sup id="cite_ref-beamer_10-2" class="reference"><a href="#cite_note-beamer-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {p\cdot q}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>⋅</mo> <mi>q</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {p\cdot q}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b248ba124ca6090b6a90c9ebe085068fe9637b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.243ex; height:4.843ex;" alt="{\displaystyle A={\frac {p\cdot q}{2}}.}"></span> Alternatively, the area can be calculated by dividing the kite into two congruent triangles and applying the <a href="/wiki/Area_of_a_triangle#Using_trigonometry" title="Area of a triangle">SAS formula</a> for their area. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> are the lengths of two sides of the kite, and <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is the <a href="/wiki/Angle" title="Angle">angle</a> between, then the area is<sup id="cite_ref-crux_26-0" class="reference"><a href="#cite_note-crux-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle A=ab\cdot \sin \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle A=ab\cdot \sin \theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df1ff61741ace310af7ca670ec3717d7e97a42f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.728ex; height:2.176ex;" alt="{\displaystyle \displaystyle A=ab\cdot \sin \theta .}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Inscribed_circle">Inscribed circle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=5" title="Edit section: Inscribed circle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:304px;max-width:304px"><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Kite_inexcircles.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Kite_inexcircles.svg/300px-Kite_inexcircles.svg.png" decoding="async" width="300" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Kite_inexcircles.svg/450px-Kite_inexcircles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Kite_inexcircles.svg/600px-Kite_inexcircles.svg.png 2x" data-file-width="657" data-file-height="306" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Dart_inexcircles.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Dart_inexcircles.svg/300px-Dart_inexcircles.svg.png" decoding="async" width="300" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Dart_inexcircles.svg/450px-Dart_inexcircles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Dart_inexcircles.svg/600px-Dart_inexcircles.svg.png 2x" data-file-width="657" data-file-height="306" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Antipar_inexcircles.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Antipar_inexcircles.svg/300px-Antipar_inexcircles.svg.png" decoding="async" width="300" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Antipar_inexcircles.svg/450px-Antipar_inexcircles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Antipar_inexcircles.svg/600px-Antipar_inexcircles.svg.png 2x" data-file-width="657" data-file-height="306" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Two circles tangent to the sides and extended sides of a convex kite (top), non-convex kite (middle), and <a href="/wiki/Antiparallelogram" title="Antiparallelogram">antiparallelogram</a> (bottom). The four lines through the sides of each quadrilateral are <a href="/wiki/Bitangent" title="Bitangent">bitangents</a> of the circles.</div></div></div></div> <p>Every <i>convex</i> kite is also a <a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">tangential quadrilateral</a>, a quadrilateral that has an <a href="/wiki/Inscribed_circle" class="mw-redirect" title="Inscribed circle">inscribed circle</a>. That is, there exists a circle that is <a href="/wiki/Tangent" title="Tangent">tangent</a> to all four sides. Additionally, if a convex kite is not a rhombus, there is a circle outside the kite that is tangent to the extensions of the four sides; therefore, every convex kite that is not a rhombus is an <a href="/wiki/Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">ex-tangential quadrilateral</a>. The convex kites that are not rhombi are exactly the quadrilaterals that are both tangential and ex-tangential.<sup id="cite_ref-alsina-nelson_16-4" class="reference"><a href="#cite_note-alsina-nelson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> For every <i>concave</i> kite there exist two circles tangent to two of the sides and the extensions of the other two: one is interior to the kite and touches the two sides opposite from the concave angle, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle.<sup id="cite_ref-wheeler_27-0" class="reference"><a href="#cite_note-wheeler-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>For a convex kite with diagonal lengths <span class="mwe-math-element"><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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> and side lengths <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, the radius <span class="mwe-math-element"><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> of the inscribed circle is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\frac {pq}{2(a+b)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mi>q</mi> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {pq}{2(a+b)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4afdd3f7c4650a36f1f6b42268505a5b3ff2ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.67ex; height:5.676ex;" alt="{\displaystyle r={\frac {pq}{2(a+b)}},}"></span> and the radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> of the ex-tangential circle is<sup id="cite_ref-alsina-nelson_16-5" class="reference"><a href="#cite_note-alsina-nelson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {pq}{2|a-b|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mi>q</mi> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {pq}{2|a-b|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d30c7e2c514f9c8643f8e4d2c6ea5ddba2b793" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.307ex; height:5.676ex;" alt="{\displaystyle \rho ={\frac {pq}{2|a-b|}}.}"></span> </p><p>A tangential quadrilateral is also a kite <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> any one of the following conditions is true:<sup id="cite_ref-josefsson-when_28-0" class="reference"><a href="#cite_note-josefsson-when-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <ul><li>The area is one half the product of the <a href="/wiki/Diagonal" title="Diagonal">diagonals</a>.</li> <li>The diagonals are <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a>. (Thus the kites are exactly the quadrilaterals that are both tangential and <a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">orthodiagonal</a>.)</li> <li>The two line segments connecting opposite points of tangency have equal length.</li> <li>The <a href="/wiki/Tangential_quadrilateral#Special_line_segments" title="Tangential quadrilateral">tangent lengths</a>, distances from a point of tangency to an adjacent vertex of the quadrilateral, are equal at two opposite vertices of the quadrilateral. (At each vertex, there are two adjacent points of tangency, but they are the same distance as each other from the vertex, so each vertex has a single tangent length.)</li> <li>The two <a href="/wiki/Quadrilateral#Special_line_segments" title="Quadrilateral">bimedians</a>, line segments connecting midpoints of opposite edges, have equal length.</li> <li>The products of opposite side lengths are equal.</li> <li>The center of the incircle lies on a line of symmetry that is also a diagonal.</li></ul> <p>If the diagonals in a tangential quadrilateral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}"></span> intersect at <span class="mwe-math-element"><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/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>, and the <a href="/wiki/Incircle_and_excircles_of_a_triangle" class="mw-redirect" title="Incircle and excircles of a triangle">incircles</a> of triangles <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470fce11bdc31584916684eae3c9f8d37cddbe5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.253ex; height:2.176ex;" alt="{\displaystyle ABP}"></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 BCP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BCP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be7a305d69647164e507744e12cfb42c592c97b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.276ex; height:2.176ex;" alt="{\displaystyle BCP}"></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 CDP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>D</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CDP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b017f4a2d60dc11b3fd3f4b0ccf6fa08db1e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.436ex; height:2.176ex;" alt="{\displaystyle CDP}"></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 DAP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>A</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DAP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f72972713095b07d7c8a8ef1809161aeebd44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.413ex; height:2.176ex;" alt="{\displaystyle DAP}"></span> have radii <span class="mwe-math-element"><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"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </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/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"></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"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </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/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{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 r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}"></span>, and <span class="mwe-math-element"><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_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ff4fdbb4deb0617c2f0597ed2195e0b7b4479f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{4}}"></span> respectively, then the quadrilateral is a kite if and only if<sup id="cite_ref-josefsson-when_28-1" class="reference"><a href="#cite_note-josefsson-when-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}+r_{3}=r_{2}+r_{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}+r_{3}=r_{2}+r_{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0236d6804ca2c7caca882bd7f7ecaaa3906a6658" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.838ex; height:2.343ex;" alt="{\displaystyle r_{1}+r_{3}=r_{2}+r_{4}.}"></span> If the <a href="/wiki/Incircle_and_excircles_of_a_triangle" class="mw-redirect" title="Incircle and excircles of a triangle">excircles</a> to the same four triangles opposite the vertex <span class="mwe-math-element"><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/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> have radii <span class="mwe-math-element"><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"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </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/c1d63c96f59d98589d923c4f0b04222feaa7283e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{1}}"></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"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </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/35f571121c264178676d1df8ab899f238a39bc2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{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 R_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b0bb30b2846df2cd6cbedc7a796388e339d0fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{3}}"></span>, and <span class="mwe-math-element"><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_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e1a5e188204fb76d61e11e89567b5c922d9ab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{4}}"></span> respectively, then the quadrilateral is a kite if and only if<sup id="cite_ref-josefsson-when_28-2" class="reference"><a href="#cite_note-josefsson-when-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{1}+R_{3}=R_{2}+R_{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}+R_{3}=R_{2}+R_{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93555fc109354c3de91e4a9a356456caee831ee0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.699ex; height:2.509ex;" alt="{\displaystyle R_{1}+R_{3}=R_{2}+R_{4}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Duality">Duality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=6" title="Edit section: Duality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kite_isotrap_duality.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Kite_isotrap_duality.svg/170px-Kite_isotrap_duality.svg.png" decoding="async" width="170" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Kite_isotrap_duality.svg/255px-Kite_isotrap_duality.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Kite_isotrap_duality.svg/340px-Kite_isotrap_duality.svg.png 2x" data-file-width="396" data-file-height="486" /></a><figcaption>A kite and its dual isosceles trapezoid</figcaption></figure> <p>Kites and <a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">isosceles trapezoids</a> are dual to each other, meaning that there is a correspondence between them that reverses the dimension of their parts, taking vertices to sides and sides to vertices. From any kite, the inscribed circle is tangent to its four sides at the four vertices of an isosceles trapezoid. For any isosceles trapezoid, tangent lines to the circumscribing circle at its four vertices form the four sides of a kite. This correspondence can also be seen as an example of <a href="/wiki/Polar_reciprocation" class="mw-redirect" title="Polar reciprocation">polar reciprocation</a>, a general method for corresponding points with lines and vice versa given a fixed circle. Although they do not touch the circle, the four vertices of the kite are reciprocal in this sense to the four sides of the isosceles trapezoid.<sup id="cite_ref-robertson_29-0" class="reference"><a href="#cite_note-robertson-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The features of kites and isosceles trapezoids that correspond to each other under this duality are compared in the table below.<sup id="cite_ref-devilliers-adventures_7-4" class="reference"><a href="#cite_note-devilliers-adventures-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="text-align: center;"> <tbody><tr> <th scope="col">Isosceles trapezoid </th> <th scope="col">Kite </th></tr> <tr> <td>Two pairs of equal adjacent angles </td> <td>Two pairs of equal adjacent sides </td></tr> <tr> <td>Two equal opposite sides </td> <td>Two equal opposite angles </td></tr> <tr> <td>Two opposite sides with a shared perpendicular bisector </td> <td>Two opposite angles with a shared angle bisector </td></tr> <tr> <td>An axis of symmetry through two opposite sides </td> <td>An axis of symmetry through two opposite angles </td></tr> <tr> <td>Circumscribed circle through all vertices </td> <td>Inscribed circle tangent to all sides </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Dissection">Dissection</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=7" title="Edit section: Dissection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Equidissection" title="Equidissection">equidissection</a> problem concerns the subdivision of polygons into triangles that all have equal areas. In this context, the <i>spectrum</i> of a polygon is the set of numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> such that the polygon has an equidissection into <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> equal-area triangles. Because of its symmetry, the spectrum of a kite contains all even integers. Certain special kites also contain some odd numbers in their spectra.<sup id="cite_ref-kasimitis-stein_30-0" class="reference"><a href="#cite_note-kasimitis-stein-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-jepsen-sedberry-hoyer_31-0" class="reference"><a href="#cite_note-jepsen-sedberry-hoyer-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>Every triangle can be subdivided into three right kites meeting at the center of its inscribed circle. More generally, a method based on <a href="/wiki/Circle_packing" title="Circle packing">circle packing</a> can be used to subdivide any polygon with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> sides into <span class="mwe-math-element"><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(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span> kites, meeting edge-to-edge.<sup id="cite_ref-bern-eppstein_32-0" class="reference"><a href="#cite_note-bern-eppstein-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Tilings_and_polyhedra">Tilings and polyhedra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=8" title="Edit section: Tilings and polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:532px;max-width:532px"><div class="trow"><div class="tsingle" style="width:252px;max-width:252px"><div class="thumbimage" style="height:263px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:AnimSun2k.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/AnimSun2k.gif/250px-AnimSun2k.gif" decoding="async" width="250" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/AnimSun2k.gif/375px-AnimSun2k.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/AnimSun2k.gif/500px-AnimSun2k.gif 2x" data-file-width="1200" data-file-height="1262" /></a></span></div><div class="thumbcaption">Recursive construction of the kite and dart Penrose tiling</div></div><div class="tsingle" style="width:276px;max-width:276px"><div class="thumbimage" style="height:263px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Fractal_Penrose_kite_rosette.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Fractal_Penrose_kite_rosette.svg/274px-Fractal_Penrose_kite_rosette.svg.png" decoding="async" width="274" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Fractal_Penrose_kite_rosette.svg/411px-Fractal_Penrose_kite_rosette.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Fractal_Penrose_kite_rosette.svg/548px-Fractal_Penrose_kite_rosette.svg.png 2x" data-file-width="693" data-file-height="666" /></a></span></div><div class="thumbcaption">Fractal rosette of Penrose kites</div></div></div></div></div> <p>All kites <a href="/wiki/Tessellation" title="Tessellation">tile the plane</a> by repeated <a href="/wiki/Point_reflection" title="Point reflection">point reflection</a> around the midpoints of their edges, as do more generally all quadrilaterals.<sup id="cite_ref-schattschneider_33-0" class="reference"><a href="#cite_note-schattschneider-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Kites and darts with angles 72°, 72°, 72°, 144° and 36°, 72°, 36°, 216°, respectively, form the <a href="/wiki/Prototile" title="Prototile">prototiles</a> of one version of the <a href="/wiki/Penrose_tiling" title="Penrose tiling">Penrose tiling</a>, an <a href="/wiki/Aperiodic_tiling" title="Aperiodic tiling">aperiodic tiling</a> of the plane discovered by mathematical physicist <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a>.<sup id="cite_ref-gardner_5-1" class="reference"><a href="#cite_note-gardner-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> When a kite has angles that, at its apex and one side, sum to <span class="mwe-math-element"><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-{\tfrac {1}{n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (1-{\tfrac {1}{n}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a35b3d77f20fb022a67925aee07e32478f9e1ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.966ex; height:3.343ex;" alt="{\displaystyle \pi (1-{\tfrac {1}{n}})}"></span> for some positive integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, then scaled copies of that kite can be used to tile the plane in a <a href="/wiki/Fractal" title="Fractal">fractal</a> rosette in which successively larger rings of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> kites surround a central point.<sup id="cite_ref-fathauer_34-0" class="reference"><a href="#cite_note-fathauer-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> These rosettes can be used to study the phenomenon of inelastic collapse, in which a system of moving particles meeting in <a href="/wiki/Inelastic_collision" title="Inelastic collision">inelastic collisions</a> all coalesce at a common point.<sup id="cite_ref-chazelle-karntikoon-zheng_35-0" class="reference"><a href="#cite_note-chazelle-karntikoon-zheng-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>A kite with angles 60°, 90°, 120°, 90° can also tile the plane by repeated reflection across its edges; the resulting tessellation, the <a href="/wiki/Deltoidal_trihexagonal_tiling" class="mw-redirect" title="Deltoidal trihexagonal tiling">deltoidal trihexagonal tiling</a>, superposes a tessellation of the plane by regular hexagons and isosceles triangles.<sup id="cite_ref-alsina-nelson_16-6" class="reference"><a href="#cite_note-alsina-nelson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">deltoidal icositetrahedron</a>, <a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">deltoidal hexecontahedron</a>, and <a href="/wiki/Trapezohedron" title="Trapezohedron">trapezohedron</a> are <a href="/wiki/Polyhedra" class="mw-redirect" title="Polyhedra">polyhedra</a> with congruent kite-shaped <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a>,<sup id="cite_ref-grunbaum_36-0" class="reference"><a href="#cite_note-grunbaum-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> which can alternatively be thought of as tilings of the sphere by congruent spherical kites.<sup id="cite_ref-sakano-akama_37-0" class="reference"><a href="#cite_note-sakano-akama-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> There are infinitely many <a href="/wiki/Isohedral_figure" title="Isohedral figure">face-symmetric tilings</a> of the <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a> by kites.<sup id="cite_ref-dunham-lindgren-witte_38-0" class="reference"><a href="#cite_note-dunham-lindgren-witte-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> These polyhedra (equivalently, spherical tilings), the square and deltoidal trihexagonal tilings of the Euclidean plane, and some tilings of the hyperbolic plane are shown in the table below, labeled by <a href="/wiki/Face_configuration" class="mw-redirect" title="Face configuration">face configuration</a> (the numbers of neighbors of each of the four vertices of each tile). Some polyhedra and tilings appear twice, under two different face configurations. </p> <table class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"> <tbody><tr> <th colspan="3">Polyhedra </th> <th>Euclidean </th></tr> <tr style="text-align:center;vertical-align:top;"> <td><span typeof="mw:File"><a href="/wiki/File:Rhombicdodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/120px-Rhombicdodecahedron.jpg" decoding="async" width="120" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/180px-Rhombicdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/240px-Rhombicdodecahedron.jpg 2x" data-file-width="849" data-file-height="754" /></a></span><br /><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">V4.3.4.3</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Deltoidalicositetrahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Deltoidalicositetrahedron.jpg/120px-Deltoidalicositetrahedron.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Deltoidalicositetrahedron.jpg/180px-Deltoidalicositetrahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Deltoidalicositetrahedron.jpg/240px-Deltoidalicositetrahedron.jpg 2x" data-file-width="845" data-file-height="837" /></a></span><br /><a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">V4.3.4.4</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Deltoidalhexecontahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/120px-Deltoidalhexecontahedron.jpg" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/180px-Deltoidalhexecontahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/240px-Deltoidalhexecontahedron.jpg 2x" data-file-width="854" data-file-height="843" /></a></span><br /><a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">V4.3.4.5</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg/120px-Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg/180px-Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg/240px-Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span><br /><a href="/wiki/Deltoidal_trihexagonal_tiling" class="mw-redirect" title="Deltoidal trihexagonal tiling">V4.3.4.6</a> </td></tr> <tr> <th>Polyhedra </th> <th>Euclidean </th> <th colspan="2">Hyperbolic tilings </th></tr> <tr style="text-align:center;vertical-align:top;"> <td><span typeof="mw:File"><a href="/wiki/File:Deltoidalicositetrahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Deltoidalicositetrahedron.jpg/120px-Deltoidalicositetrahedron.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Deltoidalicositetrahedron.jpg/180px-Deltoidalicositetrahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Deltoidalicositetrahedron.jpg/240px-Deltoidalicositetrahedron.jpg 2x" data-file-width="845" data-file-height="837" /></a></span><br /><a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">V4.4.4.3</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Square_tiling_uniform_coloring_1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Square_tiling_uniform_coloring_1.png/120px-Square_tiling_uniform_coloring_1.png" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Square_tiling_uniform_coloring_1.png/180px-Square_tiling_uniform_coloring_1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Square_tiling_uniform_coloring_1.png/240px-Square_tiling_uniform_coloring_1.png 2x" data-file-width="414" data-file-height="410" /></a></span><br /><a href="/wiki/Square_tiling" title="Square tiling">V4.4.4.4</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:H2-5-4-deltoidal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/H2-5-4-deltoidal.svg/120px-H2-5-4-deltoidal.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/H2-5-4-deltoidal.svg/180px-H2-5-4-deltoidal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/H2-5-4-deltoidal.svg/240px-H2-5-4-deltoidal.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br /><a href="/wiki/Deltoidal_tetrapentagonal_tiling" class="mw-redirect" title="Deltoidal tetrapentagonal tiling">V4.4.4.5</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:H2chess_246d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/H2chess_246d.png/120px-H2chess_246d.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/H2chess_246d.png/180px-H2chess_246d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/H2chess_246d.png/240px-H2chess_246d.png 2x" data-file-width="2520" data-file-height="2520" /></a></span><br /><a href="/wiki/Deltoidal_tetrahexagonal_tiling" class="mw-redirect" title="Deltoidal tetrahexagonal tiling">V4.4.4.6</a> </td></tr> <tr> <th>Polyhedra </th> <th colspan="3">Hyperbolic tilings </th></tr> <tr style="text-align:center;vertical-align:top;"> <td><span typeof="mw:File"><a href="/wiki/File:Deltoidalhexecontahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/120px-Deltoidalhexecontahedron.jpg" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/180px-Deltoidalhexecontahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/240px-Deltoidalhexecontahedron.jpg 2x" data-file-width="854" data-file-height="843" /></a></span><br /><a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">V4.3.4.5</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:H2-5-4-deltoidal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/H2-5-4-deltoidal.svg/120px-H2-5-4-deltoidal.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/H2-5-4-deltoidal.svg/180px-H2-5-4-deltoidal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/H2-5-4-deltoidal.svg/240px-H2-5-4-deltoidal.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br /><a href="/wiki/Deltoidal_tetrapentagonal_tiling" class="mw-redirect" title="Deltoidal tetrapentagonal tiling">V4.4.4.5</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:H2-5-4-rhombic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/H2-5-4-rhombic.svg/120px-H2-5-4-rhombic.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/H2-5-4-rhombic.svg/180px-H2-5-4-rhombic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/H2-5-4-rhombic.svg/240px-H2-5-4-rhombic.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br /><a href="/wiki/Rhombic_pentapentagonal_tiling" class="mw-redirect" title="Rhombic pentapentagonal tiling">V4.5.4.5</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Deltoidal_pentahexagonal_tiling.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Deltoidal_pentahexagonal_tiling.png/120px-Deltoidal_pentahexagonal_tiling.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Deltoidal_pentahexagonal_tiling.png/180px-Deltoidal_pentahexagonal_tiling.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Deltoidal_pentahexagonal_tiling.png/240px-Deltoidal_pentahexagonal_tiling.png 2x" data-file-width="512" data-file-height="512" /></a></span><br /><a href="/wiki/Deltoidal_pentahexagonal_tiling" class="mw-redirect" title="Deltoidal pentahexagonal tiling">V4.6.4.5</a> </td></tr> <tr> <th>Euclidean </th> <th colspan="3">Hyperbolic tilings </th></tr> <tr style="text-align:center;vertical-align:top;"> <td><span typeof="mw:File"><a href="/wiki/File:Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg/120px-Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg/180px-Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg/240px-Tiling_Dual_Semiregular_V3-4-6-4_Deltoidal_Trihexagonal.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span><br /><a href="/wiki/Deltoidal_trihexagonal_tiling" class="mw-redirect" title="Deltoidal trihexagonal tiling">V4.3.4.6</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:H2chess_246d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/H2chess_246d.png/120px-H2chess_246d.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/H2chess_246d.png/180px-H2chess_246d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/H2chess_246d.png/240px-H2chess_246d.png 2x" data-file-width="2520" data-file-height="2520" /></a></span><br /><a href="/wiki/Deltoidal_tetrahexagonal_tiling" class="mw-redirect" title="Deltoidal tetrahexagonal tiling">V4.4.4.6</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Deltoidal_pentahexagonal_tiling.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Deltoidal_pentahexagonal_tiling.png/120px-Deltoidal_pentahexagonal_tiling.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Deltoidal_pentahexagonal_tiling.png/180px-Deltoidal_pentahexagonal_tiling.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Deltoidal_pentahexagonal_tiling.png/240px-Deltoidal_pentahexagonal_tiling.png 2x" data-file-width="512" data-file-height="512" /></a></span><br /><a href="/wiki/Deltoidal_pentahexagonal_tiling" class="mw-redirect" title="Deltoidal pentahexagonal tiling">V4.5.4.6</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:H2chess_266d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/H2chess_266d.png/120px-H2chess_266d.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/H2chess_266d.png/180px-H2chess_266d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/H2chess_266d.png/240px-H2chess_266d.png 2x" data-file-width="2520" data-file-height="2520" /></a></span><br /><a href="/wiki/Rhombic_hexahexagonal_tiling" class="mw-redirect" title="Rhombic hexahexagonal tiling">V4.6.4.6</a> </td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2_10-sided_dice.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/2_10-sided_dice.jpg/170px-2_10-sided_dice.jpg" decoding="async" width="170" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/2_10-sided_dice.jpg/255px-2_10-sided_dice.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/2_10-sided_dice.jpg/340px-2_10-sided_dice.jpg 2x" data-file-width="3697" data-file-height="2449" /></a><figcaption>Ten-sided dice</figcaption></figure> <p>The <a href="/wiki/Trapezohedron" title="Trapezohedron">trapezohedra</a> are another family of polyhedra that have congruent kite-shaped faces. In these polyhedra, the edges of one of the two side lengths of the kite meet at two "pole" vertices, while the edges of the other length form an equatorial zigzag path around the polyhedron. They are the <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedra</a> of the uniform <a href="/wiki/Antiprism" title="Antiprism">antiprisms</a>.<sup id="cite_ref-grunbaum_36-1" class="reference"><a href="#cite_note-grunbaum-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> A commonly seen example is the <a href="/wiki/Pentagonal_trapezohedron" title="Pentagonal trapezohedron">pentagonal trapezohedron</a>, used for ten-sided <a href="/wiki/Dice" title="Dice">dice</a>.<sup id="cite_ref-alsina-nelson_16-7" class="reference"><a href="#cite_note-alsina-nelson-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Outer_billiards">Outer billiards</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=9" title="Edit section: Outer billiards"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematician <a href="/wiki/Richard_Schwartz_(mathematician)" title="Richard Schwartz (mathematician)">Richard Schwartz</a> has studied <a href="/wiki/Outer_billiard" class="mw-redirect" title="Outer billiard">outer billiards</a> on kites. Outer billiards is a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a> in which, from a point outside a given <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact</a> <a href="/wiki/Convex_set" title="Convex set">convex set</a> in the plane, one draws a tangent line to the convex set, travels from the starting point along this line to another point equally far from the point of tangency, and then repeats the same process. It had been open since the 1950s whether any system defined in this way could produce paths that get arbitrarily far from their starting point, and in a 2007 paper Schwartz solved this problem by finding unbounded billiards paths for the kite with angles 72°, 72°, 72°, 144°, the same as the one used in the Penrose tiling.<sup id="cite_ref-schwartz-unbounded_39-0" class="reference"><a href="#cite_note-schwartz-unbounded-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> He later wrote a <a href="/wiki/Monograph" title="Monograph">monograph</a> analyzing outer billiards for kite shapes more generally. For this problem, any <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> of a kite preserves the dynamical properties of outer billiards on it, and it is possible to transform any kite into a shape where three vertices are at the points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f5f5f03e9f4380e41314d8c5d0129861c8ecf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,0)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\pm 1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,\pm 1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33246c0d5176c84821cf5cee7b095a7a29cbbdbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (0,\pm 1)}"></span>, with the fourth at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\alpha ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b82fa6bda9220e4eafb5b46db6584a4a891aaeb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.493ex; height:2.843ex;" alt="{\displaystyle (\alpha ,0)}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> in the open unit interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>. The behavior of outer billiards on any kite depends strongly on the parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and in particular whether it is <a href="/wiki/Rational_number" title="Rational number">rational</a>. For the case of the Penrose kite, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1/\varphi ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =1/\varphi ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fdd049608d8948eeb546b4d537bc2e5bf1d2b9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.485ex; height:3.176ex;" alt="{\displaystyle \alpha =1/\varphi ^{3}}"></span>, an irrational number, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =(1+{\sqrt {5}})/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =(1+{\sqrt {5}})/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f409bafc8166035e6535ed6bb1a12ccb2d97d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.854ex; height:3.009ex;" alt="{\displaystyle \varphi =(1+{\sqrt {5}})/2}"></span> is the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>.<sup id="cite_ref-schwartz-monograph_40-0" class="reference"><a href="#cite_note-schwartz-monograph-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.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 mw-references-columns"><ol class="references"> <li id="cite_note-halsted-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-halsted_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-halsted_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-halsted_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-halsted_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-halsted_1-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHalsted1896" class="citation cs2"><a href="/wiki/G._B._Halsted" title="G. B. Halsted">Halsted, George Bruce</a> (1896), <a rel="nofollow" class="external text" href="https://archive.org/details/elementarysynth00halsgoog/page/n64">"Chapter XIV. Symmetrical Quadrilaterals"</a>, <i>Elementary Synthetic Geometry</i>, J. Wiley & sons, pp. 49–53</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+XIV.+Symmetrical+Quadrilaterals&rft.btitle=Elementary+Synthetic+Geometry&rft.pages=49-53&rft.pub=J.+Wiley+%26+sons&rft.date=1896&rft.aulast=Halsted&rft.aufirst=George+Bruce&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementarysynth00halsgoog%2Fpage%2Fn64&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-goormaghtigh-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-goormaghtigh_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoormaghtigh1947" class="citation cs2"><a href="/wiki/Ren%C3%A9_Goormaghtigh" title="René Goormaghtigh">Goormaghtigh, R.</a> (1947), "Orthopolar and isopolar lines in the cyclic quadrilateral", <i><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>, <b>54</b> (4): 211–214, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.1947.11991815">10.1080/00029890.1947.11991815</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2304700">2304700</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0019934">0019934</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Orthopolar+and+isopolar+lines+in+the+cyclic+quadrilateral&rft.volume=54&rft.issue=4&rft.pages=211-214&rft.date=1947&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D19934%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2304700%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1080%2F00029890.1947.11991815&rft.aulast=Goormaghtigh&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></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">See <a href="/wiki/H._S._M._Coxeter" class="mw-redirect" title="H. S. M. Coxeter">H. S. M. Coxeter</a>'s review of <a href="#CITEREFGrünbaum1960">Grünbaum (1960)</a> in <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0125489">0125489</a>: "It is unfortunate that the author uses, instead of 'kite', the name 'deltoid', which belongs more properly to a curve, the three-cusped hypocycloid."</span> </li> <li id="cite_note-charter-rogers-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-charter-rogers_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCharterRogers1993" class="citation cs2">Charter, Kevin; Rogers, Thomas (1993), <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.em/1062620831">"The dynamics of quadrilateral folding"</a>, <i>Experimental Mathematics</i>, <b>2</b> (3): 209–222, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10586458.1993.10504278">10.1080/10586458.1993.10504278</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1273409">1273409</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Experimental+Mathematics&rft.atitle=The+dynamics+of+quadrilateral+folding&rft.volume=2&rft.issue=3&rft.pages=209-222&rft.date=1993&rft_id=info%3Adoi%2F10.1080%2F10586458.1993.10504278&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1273409%23id-name%3DMR&rft.aulast=Charter&rft.aufirst=Kevin&rft.au=Rogers%2C+Thomas&rft_id=https%3A%2F%2Fprojecteuclid.org%2Feuclid.em%2F1062620831&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-gardner-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-gardner_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-gardner_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner1977" class="citation cs2"><a href="/wiki/Martin_Gardner" title="Martin Gardner">Gardner, Martin</a> (January 1977), "Extraordinary nonperiodic tiling that enriches the theory of tiles", Mathematical Games, <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>, vol. 236, no. 1, pp. 110–121, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1977SciAm.236a.110G">1977SciAm.236a.110G</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0177-110">10.1038/scientificamerican0177-110</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24953856">24953856</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Extraordinary+nonperiodic+tiling+that+enriches+the+theory+of+tiles&rft.volume=236&rft.issue=1&rft.pages=110-121&rft.date=1977-01&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24953856%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0177-110&rft_id=info%3Abibcode%2F1977SciAm.236a.110G&rft.aulast=Gardner&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-thurston-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-thurston_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThurston1998" class="citation cs2"><a href="/wiki/William_Thurston" title="William Thurston">Thurston, William P.</a> (1998), "Shapes of polyhedra and triangulations of the sphere", in <a href="/wiki/Igor_Rivin" title="Igor Rivin">Rivin, Igor</a>; Rourke, Colin; <a href="/wiki/Caroline_Series" title="Caroline Series">Series, Caroline</a> (eds.), <i>The Epstein birthday schrift</i>, Geometry & Topology Monographs, vol. 1, Coventry, pp. 511–549, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9801088">math/9801088</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fgtm.1998.1.511">10.2140/gtm.1998.1.511</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1668340">1668340</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8686884">8686884</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Shapes+of+polyhedra+and+triangulations+of+the+sphere&rft.btitle=The+Epstein+birthday+schrift&rft.place=Coventry&rft.series=Geometry+%26+Topology+Monographs&rft.pages=511-549&rft.date=1998&rft_id=info%3Aarxiv%2Fmath%2F9801088&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1668340%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8686884%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2140%2Fgtm.1998.1.511&rft.aulast=Thurston&rft.aufirst=William+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></span> </li> <li id="cite_note-devilliers-adventures-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-devilliers-adventures_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-devilliers-adventures_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-devilliers-adventures_7-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-devilliers-adventures_7-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-devilliers-adventures_7-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe_Villiers2009" class="citation cs2">De Villiers, Michael (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=R7uCEqwsN40C"><i>Some Adventures in Euclidean Geometry</i></a>, Dynamic Mathematics Learning, pp. 16, 55, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-557-10295-2" title="Special:BookSources/978-0-557-10295-2"><bdi>978-0-557-10295-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Some+Adventures+in+Euclidean+Geometry&rft.pages=16%2C+55&rft.pub=Dynamic+Mathematics+Learning&rft.date=2009&rft.isbn=978-0-557-10295-2&rft.aulast=De+Villiers&rft.aufirst=Michael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DR7uCEqwsN40C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-idiot-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-idiot_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSzecsei2004" class="citation cs2">Szecsei, Denise (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LAqMpvEeu5cC&pg=PA290"><i>The Complete Idiot's Guide to Geometry</i></a>, Penguin, pp. 290–291, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781592571833" title="Special:BookSources/9781592571833"><bdi>9781592571833</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Complete+Idiot%27s+Guide+to+Geometry&rft.pages=290-291&rft.pub=Penguin&rft.date=2004&rft.isbn=9781592571833&rft.aulast=Szecsei&rft.aufirst=Denise&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLAqMpvEeu5cC%26pg%3DPA290&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-usiskin-griffin-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-usiskin-griffin_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-usiskin-griffin_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUsiskinGriffin2008" class="citation cs2">Usiskin, Zalman; Griffin, Jennifer (2008), <i>The Classification of Quadrilaterals: A Study of Definition</i>, <a href="/wiki/Information_Age_Publishing" title="Information Age Publishing">Information Age Publishing</a>, pp. 49–52, 63–67</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Classification+of+Quadrilaterals%3A+A+Study+of+Definition&rft.pages=49-52%2C+63-67&rft.pub=Information+Age+Publishing&rft.date=2008&rft.aulast=Usiskin&rft.aufirst=Zalman&rft.au=Griffin%2C+Jennifer&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-beamer-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-beamer_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-beamer_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-beamer_10-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeamer1975" class="citation cs2">Beamer, James E. (May 1975), "The tale of a kite", <i>The Arithmetic Teacher</i>, <b>22</b> (5): 382–386, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2Fat.22.5.0382">10.5951/at.22.5.0382</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41188788">41188788</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Arithmetic+Teacher&rft.atitle=The+tale+of+a+kite&rft.volume=22&rft.issue=5&rft.pages=382-386&rft.date=1975-05&rft_id=info%3Adoi%2F10.5951%2Fat.22.5.0382&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F41188788%23id-name%3DJSTOR&rft.aulast=Beamer&rft.aufirst=James+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-alexander-koeberlein-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-alexander-koeberlein_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-alexander-koeberlein_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexanderKoeberlein2014" class="citation cs2">Alexander, Daniel C.; Koeberlein, Geralyn M. (2014), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EN_KAgAAQBAJ&pg=PA180"><i>Elementary Geometry for College Students</i></a> (6th ed.), <a href="/wiki/Cengage_Learning" class="mw-redirect" title="Cengage Learning">Cengage Learning</a>, pp. 180–181, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781285965901" title="Special:BookSources/9781285965901"><bdi>9781285965901</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Geometry+for+College+Students&rft.pages=180-181&rft.edition=6th&rft.pub=Cengage+Learning&rft.date=2014&rft.isbn=9781285965901&rft.aulast=Alexander&rft.aufirst=Daniel+C.&rft.au=Koeberlein%2C+Geralyn+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEN_KAgAAQBAJ%26pg%3DPA180&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-nuncius-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-nuncius_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuayTeira2014" class="citation cs2">Suay, Juan Miguel; Teira, David (2014), <a rel="nofollow" class="external text" href="https://philsci-archive.pitt.edu/18148/1/KitesFinalVersion.pdf">"Kites: the rise and fall of a scientific object"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Nuncius_(journal)" title="Nuncius (journal)">Nuncius</a></i>, <b>29</b> (2): 439–463, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1163%2F18253911-02902004">10.1163/18253911-02902004</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuncius&rft.atitle=Kites%3A+the+rise+and+fall+of+a+scientific+object&rft.volume=29&rft.issue=2&rft.pages=439-463&rft.date=2014&rft_id=info%3Adoi%2F10.1163%2F18253911-02902004&rft.aulast=Suay&rft.aufirst=Juan+Miguel&rft.au=Teira%2C+David&rft_id=https%3A%2F%2Fphilsci-archive.pitt.edu%2F18148%2F1%2FKitesFinalVersion.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-liberman-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-liberman_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiberman2009" class="citation cs2">Liberman, Anatoly (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sMiRc-JFIfMC&pg=PA17"><i>Word Origins...And How We Know Them: Etymology for Everyone</i></a>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, p. 17, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780195387070" title="Special:BookSources/9780195387070"><bdi>9780195387070</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Word+Origins...And+How+We+Know+Them%3A+Etymology+for+Everyone&rft.pages=17&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=9780195387070&rft.aulast=Liberman&rft.aufirst=Anatoly&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsMiRc-JFIfMC%26pg%3DPA17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-henrici-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-henrici_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenrici1879" class="citation cs2"><a href="/wiki/Olaus_Henrici" title="Olaus Henrici">Henrici, Olaus</a> (1879), <a rel="nofollow" class="external text" href="https://archive.org/details/elementarygeome00henrgoog/page/n20"><i>Elementary Geometry: Congruent Figures</i></a>, Longmans, Green, p. xiv</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Geometry%3A+Congruent+Figures&rft.pages=xiv&rft.pub=Longmans%2C+Green&rft.date=1879&rft.aulast=Henrici&rft.aufirst=Olaus&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementarygeome00henrgoog%2Fpage%2Fn20&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-devilliers-role-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-devilliers-role_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-devilliers-role_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe_Villiers1994" class="citation cs2">De Villiers, Michael (February 1994), "The role and function of a hierarchical classification of quadrilaterals", <i><a href="/wiki/For_the_Learning_of_Mathematics" title="For the Learning of Mathematics">For the Learning of Mathematics</a></i>, <b>14</b> (1): 11–18, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/40248098">40248098</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=For+the+Learning+of+Mathematics&rft.atitle=The+role+and+function+of+a+hierarchical+classification+of+quadrilaterals&rft.volume=14&rft.issue=1&rft.pages=11-18&rft.date=1994-02&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F40248098%23id-name%3DJSTOR&rft.aulast=De+Villiers&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-alsina-nelson-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-alsina-nelson_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-alsina-nelson_16-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-alsina-nelson_16-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-alsina-nelson_16-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-alsina-nelson_16-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-alsina-nelson_16-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-alsina-nelson_16-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-alsina-nelson_16-7"><sup><i><b>h</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlsinaNelsen2020" class="citation cs2">Alsina, Claudi; Nelsen, Roger B. (2020), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA73">"Section 3.4: Kites"</a>, <i>A Cornucopia of Quadrilaterals</i>, The Dolciani Mathematical Expositions, vol. 55, Providence, Rhode Island: MAA Press and American Mathematical Society, pp. 73–78, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-5312-1" title="Special:BookSources/978-1-4704-5312-1"><bdi>978-1-4704-5312-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=4286138">4286138</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+3.4%3A+Kites&rft.btitle=A+Cornucopia+of+Quadrilaterals&rft.place=Providence%2C+Rhode+Island&rft.series=The+Dolciani+Mathematical+Expositions&rft.pages=73-78&rft.pub=MAA+Press+and+American+Mathematical+Society&rft.date=2020&rft.isbn=978-1-4704-5312-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D4286138%23id-name%3DMR&rft.aulast=Alsina&rft.aufirst=Claudi&rft.au=Nelsen%2C+Roger+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCGDSDwAAQBAJ%26pg%3DPA73&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span>; see also antiparallelograms, p. 212</span> </li> <li id="cite_note-gant-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-gant_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGant1944" class="citation cs2">Gant, P. (1944), "A note on quadrilaterals", <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>, <b>28</b> (278): 29–30, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3607362">10.2307/3607362</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3607362">3607362</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250436895">250436895</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=A+note+on+quadrilaterals&rft.volume=28&rft.issue=278&rft.pages=29-30&rft.date=1944&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250436895%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3607362%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3607362&rft.aulast=Gant&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-josefsson-area-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-josefsson-area_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJosefsson2012" class="citation cs2">Josefsson, Martin (2012), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161007030934/http://forumgeom.fau.edu/FG2012volume12/FGvolume12.pdf#page=241">"Maximal area of a bicentric quadrilateral"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Forum_Geometricorum" title="Forum Geometricorum">Forum Geometricorum</a></i>, <b>12</b>: 237–241, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2990945">2990945</a>, archived from <a rel="nofollow" class="external text" href="http://forumgeom.fau.edu/FG2012volume12/FGvolume12.pdf#page=241">the original</a> <span class="cs1-format">(PDF)</span> on 2016-10-07<span class="reference-accessdate">, retrieved <span class="nowrap">2014-04-11</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Forum+Geometricorum&rft.atitle=Maximal+area+of+a+bicentric+quadrilateral&rft.volume=12&rft.pages=237-241&rft.date=2012&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2990945%23id-name%3DMR&rft.aulast=Josefsson&rft.aufirst=Martin&rft_id=http%3A%2F%2Fforumgeom.fau.edu%2FFG2012volume12%2FFGvolume12.pdf%23page%3D241&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-ball-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-ball_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBall1973" class="citation cs2">Ball, D. G. (1973), "A generalisation of <span class="mwe-math-element"><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>", <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>, <b>57</b> (402): 298–303, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3616052">10.2307/3616052</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3616052">3616052</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125396664">125396664</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=A+generalisation+of+MATH+RENDER+ERROR&rft.volume=57&rft.issue=402&rft.pages=298-303&rft.date=1973&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125396664%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3616052%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3616052&rft.aulast=Ball&rft.aufirst=D.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-griffiths-culpin-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-griffiths-culpin_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffithsCulpin1975" class="citation cs2">Griffiths, David; Culpin, David (1975), "Pi-optimal polygons", <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>, <b>59</b> (409): 165–175, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3617699">10.2307/3617699</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3617699">3617699</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:126325288">126325288</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=Pi-optimal+polygons&rft.volume=59&rft.issue=409&rft.pages=165-175&rft.date=1975&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126325288%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3617699%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3617699&rft.aulast=Griffiths&rft.aufirst=David&rft.au=Culpin%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-audet-hansen-svrtan-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-audet-hansen-svrtan_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAudetHansenSvrtan2021" class="citation cs2">Audet, Charles; Hansen, Pierre; Svrtan, Dragutin (2021), "Using symbolic calculations to determine largest small polygons", <i>Journal of Global Optimization</i>, <b>81</b> (1): 261–268, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10898-020-00908-w">10.1007/s10898-020-00908-w</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=4299185">4299185</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:203042405">203042405</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Global+Optimization&rft.atitle=Using+symbolic+calculations+to+determine+largest+small+polygons&rft.volume=81&rft.issue=1&rft.pages=261-268&rft.date=2021&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D4299185%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A203042405%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10898-020-00908-w&rft.aulast=Audet&rft.aufirst=Charles&rft.au=Hansen%2C+Pierre&rft.au=Svrtan%2C+Dragutin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-darling-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-darling_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDarling2004" class="citation cs2"><a href="/wiki/David_J._Darling" title="David J. Darling">Darling, David</a> (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HrOxRdtYYaMC&pg=PA260"><i>The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes</i></a>, <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, p. 260, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471667001" title="Special:BookSources/9780471667001"><bdi>9780471667001</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Universal+Book+of+Mathematics%3A+From+Abracadabra+to+Zeno%27s+Paradoxes&rft.pages=260&rft.pub=John+Wiley+%26+Sons&rft.date=2004&rft.isbn=9780471667001&rft.aulast=Darling&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHrOxRdtYYaMC%26pg%3DPA260&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-kirby-umble-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-kirby-umble_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirbyUmble2011" class="citation cs2">Kirby, Matthew; Umble, Ronald (2011), "Edge tessellations and stamp folding puzzles", <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>, <b>84</b> (4): 283–289, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0908.3257">0908.3257</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2Fmath.mag.84.4.283">10.4169/math.mag.84.4.283</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2843659">2843659</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123579388">123579388</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Edge+tessellations+and+stamp+folding+puzzles&rft.volume=84&rft.issue=4&rft.pages=283-289&rft.date=2011&rft_id=info%3Aarxiv%2F0908.3257&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2843659%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123579388%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4169%2Fmath.mag.84.4.283&rft.aulast=Kirby&rft.aufirst=Matthew&rft.au=Umble%2C+Ronald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-smkg-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-smkg_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmithMyersKaplanGoodman-Strauss2024" class="citation cs2">Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; <a href="/wiki/Chaim_Goodman-Strauss" title="Chaim Goodman-Strauss">Goodman-Strauss, Chaim</a> (2024), "An aperiodic monotile", <i>Combinatorial Theory</i>, <b>4</b>, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2303.10798">2303.10798</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5070%2FC64163843">10.5070/C64163843</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Combinatorial+Theory&rft.atitle=An+aperiodic+monotile&rft.volume=4&rft.date=2024&rft_id=info%3Aarxiv%2F2303.10798&rft_id=info%3Adoi%2F10.5070%2FC64163843&rft.aulast=Smith&rft.aufirst=David&rft.au=Myers%2C+Joseph+Samuel&rft.au=Kaplan%2C+Craig+S.&rft.au=Goodman-Strauss%2C+Chaim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-eves-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-eves_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1995" class="citation cs2"><a href="/wiki/Howard_Eves" title="Howard Eves">Eves, Howard Whitley</a> (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=B81gnTjNazMC&pg=PA245"><i>College Geometry</i></a>, <a href="/wiki/Jones_%26_Bartlett_Learning" title="Jones & Bartlett Learning">Jones & Bartlett Learning</a>, p. 245, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780867204759" title="Special:BookSources/9780867204759"><bdi>9780867204759</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+Geometry&rft.pages=245&rft.pub=Jones+%26+Bartlett+Learning&rft.date=1995&rft.isbn=9780867204759&rft.aulast=Eves&rft.aufirst=Howard+Whitley&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DB81gnTjNazMC%26pg%3DPA245&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-crux-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-crux_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://cms.math.ca/wp-content/uploads/2021/06/CRUXv47n5-b.pdf">"OC506"</a> <span class="cs1-format">(PDF)</span>, Olympiad Corner Solutions, <i><a href="/wiki/Crux_Mathematicorum" title="Crux Mathematicorum">Crux Mathematicorum</a></i>, <b>47</b> (5): 241, May 2021</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Crux+Mathematicorum&rft.atitle=OC506&rft.volume=47&rft.issue=5&rft.pages=241&rft.date=2021-05&rft_id=https%3A%2F%2Fcms.math.ca%2Fwp-content%2Fuploads%2F2021%2F06%2FCRUXv47n5-b.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-wheeler-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-wheeler_27-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWheeler1958" class="citation cs2">Wheeler, Roger F. (1958), "Quadrilaterals", <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>, <b>42</b> (342): 275–276, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3610439">10.2307/3610439</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3610439">3610439</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250434576">250434576</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=Quadrilaterals&rft.volume=42&rft.issue=342&rft.pages=275-276&rft.date=1958&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250434576%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3610439%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3610439&rft.aulast=Wheeler&rft.aufirst=Roger+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-josefsson-when-28"><span class="mw-cite-backlink">^ <a href="#cite_ref-josefsson-when_28-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-josefsson-when_28-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-josefsson-when_28-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJosefsson2011" class="citation cs2">Josefsson, Martin (2011), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221017085304/https://forumgeom.fau.edu/FG2011volume11/FG201117.pdf">"When is a tangential quadrilateral a kite?"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Forum_Geometricorum" title="Forum Geometricorum">Forum Geometricorum</a></i>, <b>11</b>: 165–174, archived from <a rel="nofollow" class="external text" href="https://forumgeom.fau.edu/FG2011volume11/FG201117.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2022-10-17<span class="reference-accessdate">, retrieved <span class="nowrap">2022-08-29</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Forum+Geometricorum&rft.atitle=When+is+a+tangential+quadrilateral+a+kite%3F&rft.volume=11&rft.pages=165-174&rft.date=2011&rft.aulast=Josefsson&rft.aufirst=Martin&rft_id=https%3A%2F%2Fforumgeom.fau.edu%2FFG2011volume11%2FFG201117.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-robertson-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-robertson_29-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobertson1977" class="citation cs2">Robertson, S. A. (1977), "Classifying triangles and quadrilaterals", <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>, <b>61</b> (415): 38–49, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3617441">10.2307/3617441</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3617441">3617441</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125355481">125355481</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=Classifying+triangles+and+quadrilaterals&rft.volume=61&rft.issue=415&rft.pages=38-49&rft.date=1977&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125355481%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3617441%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3617441&rft.aulast=Robertson&rft.aufirst=S.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-kasimitis-stein-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-kasimitis-stein_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKasimatisStein1990" class="citation cs2"><a href="/wiki/Elaine_Kasimatis" title="Elaine Kasimatis">Kasimatis, Elaine A.</a>; <a href="/wiki/Sherman_K._Stein" title="Sherman K. Stein">Stein, Sherman K.</a> (December 1990), "Equidissections of polygons", <i><a href="/wiki/Discrete_Mathematics_(journal)" title="Discrete Mathematics (journal)">Discrete Mathematics</a></i>, <b>85</b> (3): 281–294, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2890%2990384-T">10.1016/0012-365X(90)90384-T</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1081836">1081836</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0736.05028">0736.05028</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=Equidissections+of+polygons&rft.volume=85&rft.issue=3&rft.pages=281-294&rft.date=1990-12&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0736.05028%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1081836%23id-name%3DMR&rft_id=info%3Adoi%2F10.1016%2F0012-365X%2890%2990384-T&rft.aulast=Kasimatis&rft.aufirst=Elaine+A.&rft.au=Stein%2C+Sherman+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-jepsen-sedberry-hoyer-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-jepsen-sedberry-hoyer_31-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJepsenSedberryHoyer2009" class="citation cs2">Jepsen, Charles H.; Sedberry, Trevor; Hoyer, Rolf (2009), <a rel="nofollow" class="external text" href="https://msp.org/involve/2009/2-1/involve-v2-n1-p07-s.pdf">"Equidissections of kite-shaped quadrilaterals"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Involve:_A_Journal_of_Mathematics" class="mw-redirect" title="Involve: A Journal of Mathematics">Involve: A Journal of Mathematics</a></i>, <b>2</b> (1): 89–93, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Finvolve.2009.2.89">10.2140/involve.2009.2.89</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2501347">2501347</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Involve%3A+A+Journal+of+Mathematics&rft.atitle=Equidissections+of+kite-shaped+quadrilaterals&rft.volume=2&rft.issue=1&rft.pages=89-93&rft.date=2009&rft_id=info%3Adoi%2F10.2140%2Finvolve.2009.2.89&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2501347%23id-name%3DMR&rft.aulast=Jepsen&rft.aufirst=Charles+H.&rft.au=Sedberry%2C+Trevor&rft.au=Hoyer%2C+Rolf&rft_id=https%3A%2F%2Fmsp.org%2Finvolve%2F2009%2F2-1%2Finvolve-v2-n1-p07-s.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-bern-eppstein-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-bern-eppstein_32-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernEppstein2000" class="citation cs2">Bern, Marshall; <a href="/wiki/David_Eppstein" title="David Eppstein">Eppstein, David</a> (2000), "Quadrilateral meshing by circle packing", <i><a href="/wiki/International_Journal_of_Computational_Geometry_and_Applications" title="International Journal of Computational Geometry and Applications">International Journal of Computational Geometry and Applications</a></i>, <b>10</b> (4): 347–360, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cs.CG/9908016">cs.CG/9908016</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0218195900000206">10.1142/S0218195900000206</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1791192">1791192</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12228995">12228995</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Computational+Geometry+and+Applications&rft.atitle=Quadrilateral+meshing+by+circle+packing&rft.volume=10&rft.issue=4&rft.pages=347-360&rft.date=2000&rft_id=info%3Aarxiv%2Fcs.CG%2F9908016&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1791192%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12228995%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS0218195900000206&rft.aulast=Bern&rft.aufirst=Marshall&rft.au=Eppstein%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-schattschneider-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-schattschneider_33-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchattschneider1993" class="citation cs2"><a href="/wiki/Doris_Schattschneider" title="Doris Schattschneider">Schattschneider, Doris</a> (1993), <a rel="nofollow" class="external text" href="https://muse.jhu.edu/article/607100">"The fascination of tiling"</a>, in Emmer, Michele (ed.), <i>The Visual Mind: Art and Mathematics</i>, Leonardo Book Series, Cambridge, Massachusetts: <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>, pp. 157–164, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-05048-X" title="Special:BookSources/0-262-05048-X"><bdi>0-262-05048-X</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1255846">1255846</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+fascination+of+tiling&rft.btitle=The+Visual+Mind%3A+Art+and+Mathematics&rft.place=Cambridge%2C+Massachusetts&rft.series=Leonardo+Book+Series&rft.pages=157-164&rft.pub=MIT+Press&rft.date=1993&rft.isbn=0-262-05048-X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1255846%23id-name%3DMR&rft.aulast=Schattschneider&rft.aufirst=Doris&rft_id=https%3A%2F%2Fmuse.jhu.edu%2Farticle%2F607100&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-fathauer-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-fathauer_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFathauer2018" class="citation cs2">Fathauer, Robert (2018), <a rel="nofollow" class="external text" href="https://archive.bridgesmathart.org/2018/bridges2018-15.html">"Art and recreational math based on kite-tiling rosettes"</a>, in <a href="/wiki/Eve_Torrence" title="Eve Torrence">Torrence, Eve</a>; Torrence, Bruce; <a href="/wiki/Carlo_H._S%C3%A9quin" title="Carlo H. Séquin">Séquin, Carlo</a>; Fenyvesi, Kristóf (eds.), <i>Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture</i>, Phoenix, Arizona: Tessellations Publishing, pp. 15–22, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-938664-27-4" title="Special:BookSources/978-1-938664-27-4"><bdi>978-1-938664-27-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Art+and+recreational+math+based+on+kite-tiling+rosettes&rft.btitle=Proceedings+of+Bridges+2018%3A+Mathematics%2C+Art%2C+Music%2C+Architecture%2C+Education%2C+Culture&rft.place=Phoenix%2C+Arizona&rft.pages=15-22&rft.pub=Tessellations+Publishing&rft.date=2018&rft.isbn=978-1-938664-27-4&rft.aulast=Fathauer&rft.aufirst=Robert&rft_id=https%3A%2F%2Farchive.bridgesmathart.org%2F2018%2Fbridges2018-15.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-chazelle-karntikoon-zheng-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-chazelle-karntikoon-zheng_35-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChazelleKarntikoonZheng2022" class="citation cs2"><a href="/wiki/Bernard_Chazelle" title="Bernard Chazelle">Chazelle, Bernard</a>; Karntikoon, Kritkorn; Zheng, Yufei (2022), "A geometric approach to inelastic collapse", <i><a href="/wiki/Journal_of_Computational_Geometry" title="Journal of Computational Geometry">Journal of Computational Geometry</a></i>, <b>13</b> (1): 197–203, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.20382%2Fjocg.v13i1a7">10.20382/jocg.v13i1a7</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=4414332">4414332</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Geometry&rft.atitle=A+geometric+approach+to+inelastic+collapse&rft.volume=13&rft.issue=1&rft.pages=197-203&rft.date=2022&rft_id=info%3Adoi%2F10.20382%2Fjocg.v13i1a7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D4414332%23id-name%3DMR&rft.aulast=Chazelle&rft.aufirst=Bernard&rft.au=Karntikoon%2C+Kritkorn&rft.au=Zheng%2C+Yufei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-grunbaum-36"><span class="mw-cite-backlink">^ <a href="#cite_ref-grunbaum_36-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-grunbaum_36-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrünbaum1960" class="citation cs2"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, B.</a> (1960), "On polyhedra in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab83515bfb94cf95187779f517841793ac61ee5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.848ex; height:2.676ex;" alt="{\displaystyle E^{3}}"></span> having all faces congruent", <i>Bulletin of the Research Council of Israel</i>, <b>8F</b>: 215–218 (1960), <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0125489">0125489</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+Research+Council+of+Israel&rft.atitle=On+polyhedra+in+MATH+RENDER+ERROR+having+all+faces+congruent&rft.volume=8F&rft.pages=215-218+%281960%29&rft.date=1960&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D125489%23id-name%3DMR&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-sakano-akama-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-sakano-akama_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSakanoAkama2015" class="citation cs2">Sakano, Yudai; Akama, Yohji (2015), <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.hmj/1448323768">"Anisohedral spherical triangles and classification of spherical tilings by congruent kites, darts and rhombi"</a>, <i><a href="/wiki/Hiroshima_Mathematical_Journal" title="Hiroshima Mathematical Journal">Hiroshima Mathematical Journal</a></i>, <b>45</b> (3): 309–339, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.32917%2Fhmj%2F1448323768">10.32917/hmj/1448323768</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3429167">3429167</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123859584">123859584</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Hiroshima+Mathematical+Journal&rft.atitle=Anisohedral+spherical+triangles+and+classification+of+spherical+tilings+by+congruent+kites%2C+darts+and+rhombi&rft.volume=45&rft.issue=3&rft.pages=309-339&rft.date=2015&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3429167%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123859584%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.32917%2Fhmj%2F1448323768&rft.aulast=Sakano&rft.aufirst=Yudai&rft.au=Akama%2C+Yohji&rft_id=https%3A%2F%2Fprojecteuclid.org%2Feuclid.hmj%2F1448323768&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-dunham-lindgren-witte-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-dunham-lindgren-witte_38-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunhamLindgrenWitte1981" class="citation cs2">Dunham, Douglas; Lindgren, John; Witte, Dave (1981), "Creating repeating hyperbolic patterns", in Green, Doug; Lucido, Tony; <a href="/wiki/Henry_Fuchs" title="Henry Fuchs">Fuchs, Henry</a> (eds.), <i>Proceedings of the 8th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1981, Dallas, Texas, USA, August 3–7, 1981</i>, <a href="/wiki/Association_for_Computing_Machinery" title="Association for Computing Machinery">Association for Computing Machinery</a>, pp. 215–223, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F800224.806808">10.1145/800224.806808</a></span>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89791-045-1" title="Special:BookSources/0-89791-045-1"><bdi>0-89791-045-1</bdi></a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2255628">2255628</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Creating+repeating+hyperbolic+patterns&rft.btitle=Proceedings+of+the+8th+Annual+Conference+on+Computer+Graphics+and+Interactive+Techniques%2C+SIGGRAPH+1981%2C+Dallas%2C+Texas%2C+USA%2C+August+3%E2%80%937%2C+1981&rft.pages=215-223&rft.pub=Association+for+Computing+Machinery&rft.date=1981&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2255628%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F800224.806808&rft.isbn=0-89791-045-1&rft.aulast=Dunham&rft.aufirst=Douglas&rft.au=Lindgren%2C+John&rft.au=Witte%2C+Dave&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-schwartz-unbounded-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-schwartz-unbounded_39-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartz2007" class="citation cs2"><a href="/wiki/Richard_Schwartz_(mathematician)" title="Richard Schwartz (mathematician)">Schwartz, Richard Evan</a> (2007), "Unbounded orbits for outer billiards, I", <i><a href="/wiki/Journal_of_Modern_Dynamics" title="Journal of Modern Dynamics">Journal of Modern Dynamics</a></i>, <b>1</b> (3): 371–424, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0702073">math/0702073</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3934%2Fjmd.2007.1.371">10.3934/jmd.2007.1.371</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2318496">2318496</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119146537">119146537</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Modern+Dynamics&rft.atitle=Unbounded+orbits+for+outer+billiards%2C+I&rft.volume=1&rft.issue=3&rft.pages=371-424&rft.date=2007&rft_id=info%3Aarxiv%2Fmath%2F0702073&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2318496%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119146537%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.3934%2Fjmd.2007.1.371&rft.aulast=Schwartz&rft.aufirst=Richard+Evan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-schwartz-monograph-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-schwartz-monograph_40-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartz2009" class="citation cs2"><a href="/wiki/Richard_Schwartz_(mathematician)" title="Richard Schwartz (mathematician)">Schwartz, Richard Evan</a> (2009), <i>Outer Billiards on Kites</i>, Annals of Mathematics Studies, vol. 171, Princeton, New Jersey: <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9781400831975">10.1515/9781400831975</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14249-4" title="Special:BookSources/978-0-691-14249-4"><bdi>978-0-691-14249-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2562898">2562898</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Outer+Billiards+on+Kites&rft.place=Princeton%2C+New+Jersey&rft.series=Annals+of+Mathematics+Studies&rft.pub=Princeton+University+Press&rft.date=2009&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2562898%23id-name%3DMR&rft_id=info%3Adoi%2F10.1515%2F9781400831975&rft.isbn=978-0-691-14249-4&rft.aulast=Schwartz&rft.aufirst=Richard+Evan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kite_(geometry)&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.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:var(--background-color-interactive-subtle,#f8f9fa);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;min-width:0}}@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><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.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/en/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/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/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">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Deltoids" class="extiw" title="commons:Category:Deltoids">Deltoids</a></span>.</div></div> </div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Kite"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs2"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Kite.html">"Kite"</a>, <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Kite&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FKite.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKite+%28geometry%29" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/kitearea.html">area formulae</a> with interactive animation at Mathopenref.com</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Polygons_(List)" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Polygons" title="Template:Polygons"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polygons" title="Template talk:Polygons"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polygons" title="Special:EditPage/Template:Polygons"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Polygons_(List)" style="font-size:114%;margin:0 4em"><a href="/wiki/Polygon" title="Polygon">Polygons</a> (<a href="/wiki/List_of_polygons" title="List of polygons">List</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Triangle" title="Triangle">Triangles</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Acute</a></li> <li><a href="/wiki/Equilateral_triangle" title="Equilateral triangle">Equilateral</a></li> <li><a href="/wiki/Ideal_triangle" title="Ideal triangle">Ideal</a></li> <li><a href="/wiki/Isosceles_triangle" title="Isosceles triangle">Isosceles</a></li> <li><a href="/wiki/Kepler_triangle" title="Kepler triangle">Kepler</a></li> <li><a href="/wiki/Acute_and_obtuse_triangles" title="Acute and obtuse triangles">Obtuse</a></li> <li><a href="/wiki/Right_triangle" title="Right triangle">Right</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilaterals</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiparallelogram" title="Antiparallelogram">Antiparallelogram</a></li> <li><a href="/wiki/Bicentric_quadrilateral" title="Bicentric quadrilateral">Bicentric</a></li> <li><a href="/wiki/Crossed_quadrilateral" class="mw-redirect" title="Crossed quadrilateral">Crossed</a></li> <li><a href="/wiki/Cyclic_quadrilateral" title="Cyclic quadrilateral">Cyclic</a></li> <li><a href="/wiki/Equidiagonal_quadrilateral" title="Equidiagonal quadrilateral">Equidiagonal</a></li> <li><a href="/wiki/Ex-tangential_quadrilateral" title="Ex-tangential quadrilateral">Ex-tangential</a></li> <li><a href="/wiki/Harmonic_quadrilateral" title="Harmonic quadrilateral">Harmonic</a></li> <li><a href="/wiki/Isosceles_trapezoid" title="Isosceles trapezoid">Isosceles trapezoid</a></li> <li><a class="mw-selflink selflink">Kite</a></li> <li><a href="/wiki/Orthodiagonal_quadrilateral" title="Orthodiagonal quadrilateral">Orthodiagonal</a></li> <li><a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Right_kite" title="Right kite">Right kite</a></li> <li><a href="/wiki/Right_trapezoid" class="mw-redirect" title="Right trapezoid">Right trapezoid</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Tangential_quadrilateral" title="Tangential quadrilateral">Tangential</a></li> <li><a href="/wiki/Tangential_trapezoid" title="Tangential trapezoid">Tangential trapezoid</a></li> <li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By number <br />of sides</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">1–10 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Monogon" title="Monogon">Monogon (1)</a></li> <li><a href="/wiki/Digon" title="Digon">Digon (2)</a></li> <li><a href="/wiki/Triangle" title="Triangle">Triangle (3)</a></li> <li><a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral (4)</a></li> <li><a href="/wiki/Pentagon" title="Pentagon">Pentagon (5)</a></li> <li><a href="/wiki/Hexagon" title="Hexagon">Hexagon (6)</a></li> <li><a href="/wiki/Heptagon" title="Heptagon">Heptagon (7)</a></li> <li><a href="/wiki/Octagon" title="Octagon">Octagon (8)</a></li> <li><a href="/wiki/Nonagon" title="Nonagon">Nonagon/Enneagon (9)</a></li> <li><a href="/wiki/Decagon" title="Decagon">Decagon (10)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">11–20 sides</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hendecagon" title="Hendecagon">Hendecagon (11)</a></li> <li><a href="/wiki/Dodecagon" title="Dodecagon">Dodecagon (12)</a></li> <li><a href="/wiki/Tridecagon" title="Tridecagon">Tridecagon (13)</a></li> <li><a href="/wiki/Tetradecagon" title="Tetradecagon">Tetradecagon (14)</a></li> <li><a href="/wiki/Pentadecagon" title="Pentadecagon">Pentadecagon (15)</a></li> <li><a href="/wiki/Hexadecagon" title="Hexadecagon">Hexadecagon (16)</a></li> <li><a href="/wiki/Heptadecagon" title="Heptadecagon">Heptadecagon (17)</a></li> <li><a href="/wiki/Octadecagon" title="Octadecagon">Octadecagon (18)</a></li> <li><a href="/wiki/Icosagon" title="Icosagon">Icosagon (20)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">>20 sides</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Icositrigon" title="Icositrigon">Icositrigon (23)</a></li> <li><a href="/wiki/Icositetragon" title="Icositetragon">Icositetragon (24)</a></li> <li><a href="/wiki/Triacontagon" title="Triacontagon">Triacontagon (30)</a></li> <li><a href="/wiki/257-gon" title="257-gon">257-gon</a></li> <li><a href="/wiki/Chiliagon" title="Chiliagon">Chiliagon (1000)</a></li> <li><a href="/wiki/Myriagon" title="Myriagon">Myriagon (10,000)</a></li> <li><a href="/wiki/65537-gon" title="65537-gon">65537-gon</a></li> <li><a href="/wiki/Megagon" title="Megagon">Megagon (1,000,000)</a></li> <li><a href="/wiki/Apeirogon" title="Apeirogon">Apeirogon (∞)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Star_polygon" title="Star polygon">Star polygons</a><br /></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentagram" title="Pentagram">Pentagram</a></li> <li><a href="/wiki/Hexagram" title="Hexagram">Hexagram</a></li> <li><a href="/wiki/Heptagram" title="Heptagram">Heptagram</a></li> <li><a href="/wiki/Octagram" title="Octagram">Octagram</a></li> <li><a href="/wiki/Enneagram_(geometry)" title="Enneagram (geometry)">Enneagram</a></li> <li><a href="/wiki/Decagram_(geometry)" title="Decagram (geometry)">Decagram</a></li> <li><a href="/wiki/Hendecagram" title="Hendecagram">Hendecagram</a></li> <li><a href="/wiki/Dodecagram" title="Dodecagram">Dodecagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Concave_polygon" title="Concave polygon">Concave</a></li> <li><a href="/wiki/Convex_polygon" title="Convex polygon">Convex</a></li> <li><a href="/wiki/Cyclic_polygon" class="mw-redirect" title="Cyclic polygon">Cyclic</a></li> <li><a href="/wiki/Equiangular_polygon" title="Equiangular polygon">Equiangular</a></li> <li><a href="/wiki/Equilateral_polygon" title="Equilateral polygon">Equilateral</a></li> <li><a href="/wiki/Infinite_skew_polygon" title="Infinite skew polygon">Infinite skew</a></li> <li><a href="/wiki/Isogonal_figure" title="Isogonal figure">Isogonal</a></li> <li><a href="/wiki/Isotoxal_figure" title="Isotoxal figure">Isotoxal</a></li> <li><a href="/wiki/Magic_polygon" title="Magic polygon">Magic</a></li> <li><a href="/wiki/Pseudotriangle" title="Pseudotriangle">Pseudotriangle</a></li> <li><a href="/wiki/Rectilinear_polygon" title="Rectilinear polygon">Rectilinear</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">Regular</a></li> <li><a href="/wiki/Reinhardt_polygon" title="Reinhardt polygon">Reinhardt</a></li> <li><a href="/wiki/Simple_polygon" title="Simple polygon">Simple</a></li> <li><a href="/wiki/Skew_polygon" title="Skew polygon">Skew</a></li> <li><a href="/wiki/Star-shaped_polygon" title="Star-shaped polygon">Star-shaped</a></li> <li><a href="/wiki/Tangential_polygon" title="Tangential polygon">Tangential</a></li> <li><a href="/wiki/Weakly_simple_polygon" title="Weakly simple polygon">Weakly simple</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Kite_(geometry)&oldid=1257445490">https://en.wikipedia.org/w/index.php?title=Kite_(geometry)&oldid=1257445490</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Elementary_shapes" title="Category:Elementary shapes">Elementary shapes</a></li><li><a href="/wiki/Category:Types_of_quadrilaterals" title="Category:Types of quadrilaterals">Types of quadrilaterals</a></li><li><a href="/wiki/Category:Kites" title="Category:Kites">Kites</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">CS1 maint: location missing publisher</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Good_articles" title="Category:Good articles">Good articles</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_October_2022" title="Category:Use dmy dates from October 2022">Use dmy dates from October 2022</a></li><li><a href="/wiki/Category:Use_list-defined_references_from_October_2022" title="Category:Use list-defined references from October 2022">Use list-defined references from October 2022</a></li><li><a href="/wiki/Category:Pages_using_multiple_image_with_auto_scaled_images" title="Category:Pages using multiple image with auto scaled images">Pages using multiple image with auto scaled images</a></li><li><a href="/wiki/Category:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category link is on Wikidata</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"> This page was last edited on 14 November 2024, at 22:57<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Kite_(geometry)&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6d67bf974d-sm5pv","wgBackendResponseTime":142,"wgPageParseReport":{"limitreport":{"cputime":"1.128","walltime":"1.394","ppvisitednodes":{"value":17736,"limit":1000000},"postexpandincludesize":{"value":127780,"limit":2097152},"templateargumentsize":{"value":4631,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":8,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":176636,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1132.676 1 -total"," 41.68% 472.142 1 Template:Reflist"," 29.35% 332.453 39 Template:Citation"," 21.84% 247.362 55 Template:R"," 20.36% 230.610 63 Template:R/ref"," 9.14% 103.580 1 Template:Polygons"," 9.10% 103.122 2 Template:Navbox"," 8.78% 99.478 1 Template:Short_description"," 6.46% 73.201 63 Template:R/superscript"," 6.34% 71.775 1 Template:Commons_category"]},"scribunto":{"limitreport-timeusage":{"value":"0.571","limit":"10.000"},"limitreport-memusage":{"value":8533479,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAlexanderKoeberlein2014\"] = 1,\n [\"CITEREFAlsinaNelsen2020\"] = 1,\n [\"CITEREFAudetHansenSvrtan2021\"] = 1,\n [\"CITEREFBall1973\"] = 1,\n [\"CITEREFBeamer1975\"] = 1,\n [\"CITEREFBernEppstein2000\"] = 1,\n [\"CITEREFCharterRogers1993\"] = 1,\n [\"CITEREFChazelleKarntikoonZheng2022\"] = 1,\n [\"CITEREFDarling2004\"] = 1,\n [\"CITEREFDe_Villiers1994\"] = 1,\n [\"CITEREFDe_Villiers2009\"] = 1,\n [\"CITEREFDunhamLindgrenWitte1981\"] = 1,\n [\"CITEREFEves1995\"] = 1,\n [\"CITEREFFathauer2018\"] = 1,\n [\"CITEREFGant1944\"] = 1,\n [\"CITEREFGardner1977\"] = 1,\n [\"CITEREFGoormaghtigh1947\"] = 1,\n [\"CITEREFGriffithsCulpin1975\"] = 1,\n [\"CITEREFGrünbaum1960\"] = 1,\n [\"CITEREFHalsted1896\"] = 1,\n [\"CITEREFHenrici1879\"] = 1,\n [\"CITEREFJepsenSedberryHoyer2009\"] = 1,\n [\"CITEREFJosefsson2011\"] = 1,\n [\"CITEREFJosefsson2012\"] = 1,\n [\"CITEREFKasimatisStein1990\"] = 1,\n [\"CITEREFKirbyUmble2011\"] = 1,\n [\"CITEREFLiberman2009\"] = 1,\n [\"CITEREFRobertson1977\"] = 1,\n [\"CITEREFSakanoAkama2015\"] = 1,\n [\"CITEREFSchattschneider1993\"] = 1,\n [\"CITEREFSchwartz2007\"] = 1,\n [\"CITEREFSchwartz2009\"] = 1,\n [\"CITEREFSmithMyersKaplanGoodman-Strauss2024\"] = 1,\n [\"CITEREFSuayTeira2014\"] = 1,\n [\"CITEREFSzecsei2004\"] = 1,\n [\"CITEREFThurston1998\"] = 1,\n [\"CITEREFUsiskinGriffin2008\"] = 1,\n [\"CITEREFWheeler1958\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Citation\"] = 39,\n [\"Commons category\"] = 1,\n [\"Good article\"] = 1,\n [\"Harvtxt\"] = 1,\n [\"Infobox polygon\"] = 1,\n [\"MR\"] = 1,\n [\"MathWorld\"] = 1,\n [\"Multiple image\"] = 2,\n [\"Multiple_image\"] = 1,\n [\"Polygons\"] = 1,\n [\"R\"] = 55,\n [\"Reflist\"] = 1,\n [\"Short description\"] = 1,\n [\"Slink\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Use list-defined references\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-lnlsq","timestamp":"20241125133747","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Kite (geometry)","url":"https:\/\/en.wikipedia.org\/wiki\/Kite_(geometry)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q107061","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q107061","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-09-13T14:27:53Z","dateModified":"2024-11-14T22:57:27Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/80\/GeometricKite.svg","headline":"quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other"}</script> </body> </html>