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</nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Obsah" 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">Obsah</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skryť</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">Začiatok</div> </a> </li> <li id="toc-Definícia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definícia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definícia</span> </div> </a> <ul id="toc-Definícia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Príklady" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Príklady"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Príklady</span> </div> </a> <ul id="toc-Príklady-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vlastnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vlastnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Vlastnosti</span> </div> </a> <ul id="toc-Vlastnosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pozri_aj" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pozri_aj"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Pozri aj</span> </div> </a> <ul id="toc-Pozri_aj-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencie"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Referencie</span> </div> </a> <ul id="toc-Referencie-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="Obsah" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Obsah" > <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="Prepnúť obsah" > <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">Prepnúť obsah</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">Stredová súmernosť</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="Prejsť na článok v inom jazyku. Je dostupný v 21 jazykoch" > <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-21" 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">21 jazykov</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A6%D1%8D%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%81%D1%96%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%8F" title="Цэнтральная сіметрыя – bieloruština" lang="be" hreflang="be" data-title="Цэнтральная сіметрыя" data-language-autonym="Беларуская" data-language-local-name="bieloruština" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/St%C5%99edov%C3%A1_soum%C4%9Brnost" title="Středová souměrnost – čeština" lang="cs" hreflang="cs" data-title="Středová souměrnost" data-language-autonym="Čeština" data-language-local-name="čeština" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Punktsymmetrie" title="Punktsymmetrie – nemčina" lang="de" hreflang="de" data-title="Punktsymmetrie" data-language-autonym="Deutsch" data-language-local-name="nemčina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BC%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1_%CF%89%CF%82_%CF%80%CF%81%CE%BF%CF%82_%CF%83%CE%B7%CE%BC%CE%B5%CE%AF%CE%BF" title="Συμμετρία ως προς σημείο – gréčtina" lang="el" hreflang="el" data-title="Συμμετρία ως προς σημείο" data-language-autonym="Ελληνικά" data-language-local-name="gréčtina" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Point_reflection" title="Point reflection – angličtina" lang="en" hreflang="en" data-title="Point reflection" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Centra_simetrio" title="Centra simetrio – esperanto" lang="eo" hreflang="eo" data-title="Centra simetrio" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Simetr%C3%ADa_central" title="Simetría central – španielčina" lang="es" hreflang="es" data-title="Simetría central" data-language-autonym="Español" data-language-local-name="španielčina" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Simetria_zentral" title="Simetria zentral – baskičtina" lang="eu" hreflang="eu" data-title="Simetria zentral" data-language-autonym="Euskara" data-language-local-name="baskičtina" 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/%D8%A8%D8%A7%D8%B2%D8%AA%D8%A7%D8%A8_%D9%86%D9%82%D8%B7%D9%87%E2%80%8C%D8%A7%DB%8C" title="بازتاب نقطه‌ای – perzština" lang="fa" hreflang="fa" data-title="بازتاب نقطه‌ای" data-language-autonym="فارسی" data-language-local-name="perzština" 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/Sym%C3%A9trie_centrale" title="Symétrie centrale – francúzština" lang="fr" hreflang="fr" data-title="Symétrie centrale" data-language-autonym="Français" data-language-local-name="francúzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Simmetria_centrale" title="Simmetria centrale – taliančina" lang="it" hreflang="it" data-title="Simmetria centrale" data-language-autonym="Italiano" data-language-local-name="taliančina" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%82%B9%E5%AF%BE%E7%A7%B0" title="点対称 – japončina" lang="ja" hreflang="ja" data-title="点対称" data-language-autonym="日本語" data-language-local-name="japončina" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%BB%D1%96%D0%BA_%D1%81%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Центрлік симметрия – kazaština" lang="kk" hreflang="kk" data-title="Центрлік симметрия" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%90%EB%8C%80%EC%B9%AD" title="점대칭 – kórejčina" lang="ko" hreflang="ko" data-title="점대칭" data-language-autonym="한국어" data-language-local-name="kórejčina" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Symetria_%C5%9Brodkowa" title="Symetria środkowa – poľština" lang="pl" hreflang="pl" data-title="Symetria środkowa" data-language-autonym="Polski" data-language-local-name="poľština" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Simetrie_fa%C8%9B%C4%83_de_centru" title="Simetrie față de centru – rumunčina" lang="ro" hreflang="ro" data-title="Simetrie față de centru" data-language-autonym="Română" data-language-local-name="rumunčina" 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%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Центральная симметрия – ruština" lang="ru" hreflang="ru" data-title="Центральная симметрия" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%81%E0%AE%B3%E0%AF%8D%E0%AE%B3%E0%AE%BF_%E0%AE%8E%E0%AE%A4%E0%AE%BF%E0%AE%B0%E0%AF%8A%E0%AE%B3%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="புள்ளி எதிரொளிப்பு – tamilčina" lang="ta" hreflang="ta" data-title="புள்ளி எதிரொளிப்பு" data-language-autonym="தமிழ்" data-language-local-name="tamilčina" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D1%81%D0%B8%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Центральна симетрія – ukrajinčina" lang="uk" hreflang="uk" data-title="Центральна симетрія" data-language-autonym="Українська" data-language-local-name="ukrajinčina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%91i_x%E1%BB%A9ng_t%C3%A2m" title="Đối xứng tâm – vietnamčina" lang="vi" hreflang="vi" data-title="Đối xứng tâm" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamčina" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%82%B9%E5%8F%8D%E6%BC%94" title="点反演 – čínština" lang="zh" hreflang="zh" data-title="点反演" data-language-autonym="中文" data-language-local-name="čínština" 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/Q1203075#sitelinks-wikipedia" title="Upraviť medzijazykové odkazy" class="wbc-editpage">Upraviť odkazy</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="Menné priestory"> <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/Stredov%C3%A1_s%C3%BAmernos%C5%A5" title="Zobraziť obsah stránky [c]" accesskey="c"><span>Stránka</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Diskusia:Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=edit&redlink=1" rel="discussion" class="new" title="Diskusia o obsahu stránky (stránka neexistuje) [t]" accesskey="t"><span>Diskusia</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="Změnit variantu jazyka" > <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">slovenčina</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="Zobrazenia"> <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/Stredov%C3%A1_s%C3%BAmernos%C5%A5"><span>Čítať</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&veaction=edit" title="Upraviť túto stránku [v]" accesskey="v"><span>Upraviť</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=edit" title="Upraviť zdrojový kód tejto stránky [e]" accesskey="e"><span>Upraviť zdroj</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=history" title="Minulé revízie tejto stránky. [h]" accesskey="h"><span>Zobraziť históriu</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Nástroje stránky"> <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="Nástroje" > <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">Nástroje</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">Nástroje</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">skryť</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Další možnosti" > <div class="vector-menu-heading"> Akcie </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/Stredov%C3%A1_s%C3%BAmernos%C5%A5"><span>Čítať</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&veaction=edit" title="Upraviť túto stránku [v]" accesskey="v"><span>Upraviť</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=edit" title="Upraviť zdrojový kód tejto stránky [e]" accesskey="e"><span>Upraviť zdroj</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=history"><span>Zobraziť históriu</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Všeobecné </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%C5%A0peci%C3%A1lne:%C4%8CoOdkazujeSem/Stredov%C3%A1_s%C3%BAmernos%C5%A5" title="Zoznam všetkých wiki stránok, ktoré sem odkazujú [j]" accesskey="j"><span>Odkazy na túto stránku</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%C5%A0peci%C3%A1lne:S%C3%BAvisiacePosledn%C3%A9%C3%9Apravy/Stredov%C3%A1_s%C3%BAmernos%C5%A5" rel="nofollow" title="Posledné úpravy na stránkach, na ktoré odkazuje táto stránka [k]" accesskey="k"><span>Súvisiace úpravy</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=sk" title="Nahranie súborov [u]" accesskey="u"><span>Nahrať súbor</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&oldid=6413749" title="Trvalý odkaz na túto verziu stránky"><span>Trvalý odkaz</span></a></li><li id="t-info" class="mw-list-item"><a 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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">Vzhľad</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skryť</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">z Wikipédie, slobodnej encyklopédie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sk" dir="ltr"><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/S%C3%BAbor:Symetria_srodkowa1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Symetria_srodkowa1.png/220px-Symetria_srodkowa1.png" decoding="async" width="220" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/7c/Symetria_srodkowa1.png 1.5x" data-file-width="300" data-file-height="271" /></a><figcaption>Stredovo súmerný útvar</figcaption></figure> <p><b>Stredová súmernosť</b> alebo <b>zrkadlový obraz</b> určený bodom S, je také <a href="/wiki/Zhodn%C3%A9_zobrazenie" title="Zhodné zobrazenie">zhodné zobrazenie</a> v <a href="/wiki/Rovina" class="mw-disambig" title="Rovina">rovine</a>, alebo v <a href="/wiki/3D" class="mw-redirect" title="3D">trojrozmernom</a> <a href="/wiki/Priestor" class="mw-disambig" title="Priestor">priestore</a>, ktoré bodu S (nazývanému stred zobrazenia) priradí ten istý bod, a k bodu A ktorý neleží v bode S priradí bod A’, pričom zároveň platí: vzdialenosť [A, S]=[A’,S] a úsečka [A, A’] leží na priamke prechádzajúcej bodom S. <b>Stredová súmernosť</b> je typ <a href="/w/index.php?title=Geometrick%C3%A9_zobrazenie&action=edit&redlink=1" class="new" title="Geometrické zobrazenie (stránka neexistuje)">geometrického zobrazenia</a>. Stredová súmernosť zachováva <a href="/wiki/Vzdialenos%C5%A5" title="Vzdialenosť">vzdialenosti</a> a <a href="/wiki/Uhol" title="Uhol">uhly</a>, ide teda o jedno zo <a href="/wiki/Zhodn%C3%A9_zobrazenie" title="Zhodné zobrazenie">zhodných zobrazení</a> v <a href="/wiki/Rovina_(geometria)" title="Rovina (geometria)">rovine</a> (alebo <a href="/wiki/Priestor" class="mw-disambig" title="Priestor">priestore</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definícia"><span id="Defin.C3.ADcia"></span>Definícia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&veaction=edit&section=1" title="Upraviť sekciu: Definícia" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=edit&section=1" title="Editovat zdrojový kód sekce Definícia"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/S%C3%BAbor:Geom_shodnost_soumernost_stred.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Geom_shodnost_soumernost_stred.svg/250px-Geom_shodnost_soumernost_stred.svg.png" decoding="async" width="250" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Geom_shodnost_soumernost_stred.svg/375px-Geom_shodnost_soumernost_stred.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Geom_shodnost_soumernost_stred.svg/500px-Geom_shodnost_soumernost_stred.svg.png 2x" data-file-width="670" data-file-height="200" /></a><figcaption>Stredová súmernosť</figcaption></figure> <p><b>Stredová súmernosť</b> <a href="/wiki/Priamka" title="Priamka">priamky</a>, <a href="/wiki/Rovina" class="mw-disambig" title="Rovina">roviny</a> alebo priestoru so stredom v bode <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> (tzv. <b>stred súmernosti</b>) je také <a href="/w/index.php?title=Geometrick%C3%A9_zobrazenie&action=edit&redlink=1" class="new" title="Geometrické zobrazenie (stránka neexistuje)">zobrazenie</a>, ktoré zobrazuje <a href="/wiki/Stred" class="mw-disambig" title="Stred">stred</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> na seba samého a bod <span class="mwe-math-element"><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> rôzny od <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> na bod <span class="mwe-math-element"><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^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f9db61370bf443bcc8eb09aa0a01406998c42fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A^{\prime }}" /></span>, ktorý sa nachádza na <a href="/wiki/Polpriamka" title="Polpriamka">polopriamke</a> opačnej k <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62363e11b1391700cb7f5a98579613430d47fa1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.242ex; height:2.176ex;" alt="{\displaystyle SA}" /></span> v rovnakej vzdialenosti od <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> ako bod <span class="mwe-math-element"><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> (čiže pre neho platí <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |SA|=|SA^{\prime }|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |SA|=|SA^{\prime }|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372e5b2401eefbdce4eb56fc18c7a5e90713e46f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.855ex; height:3.009ex;" alt="{\displaystyle |SA|=|SA^{\prime }|}" /></span>). </p><p>Objekt (či už na priamke, v rovine alebo v priestore) označujeme za <b>stredovo súmerný</b>, pokiaľ je v nejakej stredovej súmernosti obrazom samého seba. Stred tejto stredovej súmernosti potom nazývame <b>stredom súmernosti objektu</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Príklady"><span id="Pr.C3.ADklady"></span>Príklady</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&veaction=edit&section=2" title="Upraviť sekciu: Príklady" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=edit&section=2" title="Editovat zdrojový kód sekce Príklady"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/S%C3%BAbor:Point_symmetry.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Point_symmetry.jpg/250px-Point_symmetry.jpg" decoding="async" width="220" height="59" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Point_symmetry.jpg/330px-Point_symmetry.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e5/Point_symmetry.jpg 2x" data-file-width="400" data-file-height="107" /></a><figcaption>Príklady bodovej symetrie v rovine</figcaption></figure> <ul><li><a href="/wiki/%C3%9Ase%C4%8Dka_(geometria)" title="Úsečka (geometria)">Úsečka</a> alebo <a href="/wiki/Zjednotenie" class="mw-disambig" title="Zjednotenie">zjednotenie</a> dvoch úsečiek rovnakej dĺžky je príkladom stredovo súmerných objektov na priamke. <ul><li>Oproti tomu žiadna <a href="/wiki/Polpriamka" title="Polpriamka">polpriamka</a> nie je na priamke stredovo súmerná.</li></ul></li> <li><a href="/wiki/Obd%C4%BA%C5%BEnik" title="Obdĺžnik">Obdĺžnik</a>, <a href="/wiki/%C5%A0tvorec" title="Štvorec">štvorec</a>, <a href="/wiki/Koso%C5%A1tvorec" title="Kosoštvorec">kosoštvorec</a>, pravidelný <a href="/wiki/%C5%A0es%C5%A5uholn%C3%ADk" title="Šesťuholník">šesťuholník</a> a <a href="/wiki/Kruh" title="Kruh">kruh</a> sú príkladmi stredovo súmerných geometrických útvarov v rovine. <ul><li>Oproti tomu žiaden <a href="/wiki/Mnohouholn%C3%ADk" title="Mnohouholník">mnohouholník</a> s nepárnym počtom vrcholov (napr. <a href="/wiki/Trojuholn%C3%ADk" title="Trojuholník">trojuholník</a>) nie je stredovo súmerný.</li></ul></li> <li><a href="/wiki/Hyperbola_(matematika)" title="Hyperbola (matematika)">Hyperbola</a> a <a href="/wiki/Elipsa" title="Elipsa">elipsa</a> sú stredovo súmerné rovinné útvary, ale <a href="/wiki/Parabola" title="Parabola">parabola</a> stredovo súmerná nie je.</li> <li><a href="/wiki/Kocka" title="Kocka">Kocka</a>, <a href="/wiki/Gu%C4%BEa" class="mw-disambig" title="Guľa">guľa</a>, <a href="/wiki/Valec" class="mw-disambig" title="Valec">valec</a> sú stredovo súmerné geometrické objekty v priestore. <ul><li>Naproti tomu žiaden <a href="/wiki/Ihlan" title="Ihlan">ihlan</a> ani <a href="/wiki/Ku%C5%BEe%C4%BE" title="Kužeľ">kužeľ</a> nie je stredovo súmerný.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Vlastnosti">Vlastnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&veaction=edit&section=3" title="Upraviť sekciu: Vlastnosti" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=edit&section=3" title="Editovat zdrojový kód sekce Vlastnosti"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stredová súmernosť s pevne daným stredom je sama sebe <a href="/wiki/Inverzn%C3%A9_zobrazenie" class="mw-disambig" title="Inverzné zobrazenie">inverzným zobrazením</a>. Zložením dvoch stredových súmerností s rovnakým stredom vzniká <a href="/wiki/Identita_(matematika)" class="mw-redirect" title="Identita (matematika)">identita</a>. </p><p>Okrem vzdialeností zachováva stredová súmernosť v rovine i orientáciu – pokiaľ bolo poradie vrcholov v <a href="/wiki/Trojuholn%C3%ADk" title="Trojuholník">trojuholníku</a> v smere hodinových ručičiek, potom poradie ich obrazov v stredovej súmernosti je opäť v smere hodinových ručičiek (čo napr. neplatí pre <a href="/wiki/Osov%C3%A1_s%C3%BAmernos%C5%A5" title="Osová súmernosť">osovú súmernosť</a>). </p><p>Stredová súmernosť so stredom v bode <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> je v rovine zhodná s <a href="/wiki/Rot%C3%A1cia_(geometria)" title="Rotácia (geometria)">otočením</a> o 180 stupňov podľa stredu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span>. V priestore, nemá zmysel hovoriť o otočení okolo bodu, ale iba okolo osi. </p><p>Stredová súmernosť je <a href="/wiki/Invol%C3%BAcia" class="mw-disambig" title="Involúcia">involúciou</a>, pretože bod <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> je <a href="/w/index.php?title=Samodru%C5%BEn%C3%BD_bod&action=edit&redlink=1" class="new" title="Samodružný bod (stránka neexistuje)">samodružný</a> a každá <a href="/wiki/Priamka" title="Priamka">priamka</a> prechádzajúca týmto bodom je taktiež samodružná.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Pozri_aj">Pozri aj</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&veaction=edit&section=4" title="Upraviť sekciu: Pozri aj" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=edit&section=4" title="Editovat zdrojový kód sekce Pozri aj"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Osov%C3%A1_s%C3%BAmernos%C5%A5" title="Osová súmernosť">Osová súmernosť</a></li> <li><a href="/w/index.php?title=Rovinn%C3%A1_s%C3%BAmernos%C5%A5&action=edit&redlink=1" class="new" title="Rovinná súmernosť (stránka neexistuje)">Rovinná súmernosť</a></li> <li><a href="/wiki/Zhodn%C3%A9_zobrazenie" title="Zhodné zobrazenie">Zhodné zobrazenie</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencie">Referencie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&veaction=edit&section=5" title="Upraviť sekciu: Referencie" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Stredov%C3%A1_s%C3%BAmernos%C5%A5&action=edit&section=5" title="Editovat zdrojový kód sekce Referencie"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite style="font-style:normal">J. FECENKO - Ľ. PINDA. <i>Matematika 1</i>. Bratislava: Vydavateľstvo technickej a ekonomickej literatúry, 2006, [cit. 2006-03-14]. <span style="white-space:nowrap"><a href="/wiki/%C5%A0peci%C3%A1lne:Kni%C5%BEn%C3%A9Zdroje/80-8078-091-9" title="Špeciálne:KnižnéZdroje/80-8078-091-9">ISBN <span class="ISBN">80-8078-091-9</span></a></span>. </cite></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite style="font-style:normal">M. BILLICH - M. TRENKLER. <i>Zbierka úloh z geometrie</i>. Ružomberok: Pedagogická fakulta Katolíckej univerzity, 2013, [cit. 2013-03-14]. <span style="white-space:nowrap"><a href="/wiki/%C5%A0peci%C3%A1lne:Kni%C5%BEn%C3%A9Zdroje/978-80-561-0058-5" title="Špeciálne:KnižnéZdroje/978-80-561-0058-5">ISBN <span class="ISBN">978-80-561-0058-5</span></a></span>. </cite></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite style="font-style:normal">P. HORÁK - Ľ. NIEPEL. <i>Prehľad matematiky</i>. Bratislava: Vydavateľstvo technickej a ekonomickej literatúry, 1982, [cit. 1982-03-14]. </cite></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Zdroj: „<a dir="ltr" href="https://sk.wikipedia.org/w/index.php?title=Stredová_súmernosť&oldid=6413749">https://sk.wikipedia.org/w/index.php?title=Stredová_súmernosť&oldid=6413749</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%C5%A0peci%C3%A1lne:Kateg%C3%B3rie" title="Špeciálne:Kategórie">Kategória</a>: <ul><li><a href="/wiki/Kateg%C3%B3ria:Geometria" title="Kategória:Geometria">Geometria</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"> Dátum a čas poslednej úpravy tejto stránky: 22. december 2016, 21:37.</li> <li id="footer-info-copyright">Text je dostupný za podmienok <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution/Share-Alike License 4.0 Unported</a>; prípadne za ďalších podmienok. 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