{"id":855,"date":"2026-06-28T12:41:16","date_gmt":"2026-06-28T12:41:16","guid":{"rendered":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/orthocentral-quadrilateral-a-synthetic-approach\/"},"modified":"2026-06-28T12:41:16","modified_gmt":"2026-06-28T12:41:16","slug":"orthocentral-quadrilateral-a-synthetic-approach","status":"publish","type":"post","link":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/en\/orthocentral-quadrilateral-a-synthetic-approach\/","title":{"rendered":"Orthocentral quadrilateral: a synthetic approach"},"content":{"rendered":"<p><em>Ji\u0159\u00ed Bla\u017eek<\/em><br \/><small>Technick\u00e1 univerzita v Liberci (Fakulta Pedagogick\u00e1), Liberec<\/small><\/p>\n<p>In this talk, we will present some results of the article \u201cOrthocentral Quadrilaterals and Their Properties,\u201d which the author published together with Professor Pech. However, we will attempt to provide different proofs for the claims presented here\u2014proofs that rely less on algebra and manipulations of expressions and more on synthetic methods. Specifically, we will prove the following statement using synthetic methods: Let ABCD be a quadrilateral; we construct its orthocentral quadrilateral, i.e., we construct the orthocenters of the triangles ABC, BCD, CDA, and DAB. Then the resulting quadrilateral has the same area as the quadrilateral ABCD.<\/p>\n<p>(\u010desk\u00e1 verze:)<br \/>\nV p\u0159\u00edsp\u011bvku budeme prezentovat n\u011bkter\u00e9 v\u00fdsledky \u010dl\u00e1nku \u201eOrthocentral quadrilaterals and their properties\u201c, kter\u00fd autor publikoval spole\u010dn\u011b s profesorem Pechem. Pokus\u00edme se v\u0161ak o jin\u00e9 d\u016fkazy zde uveden\u00fdch tvrzen\u00ed, kter\u00e9 tolik nez\u00e1visej\u00ed na algeb\u0159e a \u00faprav\u00e1ch v\u00fdraz\u016f, ale na syntetick\u00fdch postupech. Konkr\u00e9tn\u011b dok\u00e1\u017eeme synteticky n\u00e1sleduj\u00edc\u00ed tvrzen\u00ed: Nech\u0165 je d\u00e1n \u010dty\u0159\u00faheln\u00edk ABCD, sestroj\u00edme jeho Ortocentrick\u00fd \u010dty\u0159\u00faheln\u00edk, tzn., sestroj\u00edme ortocentra troj\u00faheln\u00edku ABC, BCD, CDA, DAB. Pak tato vznikl\u00fd \u010dty\u0159\u00faheln\u00edk m\u00e1 stejn\u00fd obsah jako \u010dty\u0159\u00faheln\u00edk ABCD.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ji\u0159\u00ed Bla\u017eekTechnick\u00e1 univerzita v Liberci (Fakulta Pedagogick\u00e1), Liberec In this talk, we will present some results of the article \u201cOrthocentral Quadrilaterals and Their Properties,\u201d which the author published together with Professor Pech. However, we will attempt to provide different proofs for the claims presented here\u2014proofs that rely less on algebra and manipulations of expressions and [&hellip;]<\/p>\n","protected":false},"author":44,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-855","post","type-post","status-publish","format-standard","hentry","category-presentations"],"_links":{"self":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts\/855","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/users\/44"}],"replies":[{"embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/comments?post=855"}],"version-history":[{"count":0,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts\/855\/revisions"}],"wp:attachment":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/media?parent=855"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/categories?post=855"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/tags?post=855"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}