{"id":28,"date":"2015-12-21T02:49:06","date_gmt":"2015-12-21T02:49:06","guid":{"rendered":"http:\/\/www.ferzkopp.org\/wordpress\/?p=28"},"modified":"2016-01-08T07:44:29","modified_gmt":"2016-01-08T15:44:29","slug":"the-schiffler-point","status":"publish","type":"post","link":"https:\/\/www.ferzkopp.net\/wordpress\/2015\/12\/21\/the-schiffler-point\/","title":{"rendered":"The Schiffler Point"},"content":{"rendered":"<p>A <em>Schiffler Point<\/em>? That is cool!&#8230; Any relationships to persons of the same name (like myself) are purely coincidental.<\/p>\n<p><!--more--><\/p>\n<p>The Schiffler Point is named for Kurt Schiffler (1896-1986), who introduced the point in a problem proposal:<\/p>\n<blockquote><p><strong>Kurt Schiffler, G. R. Veldkamp, and W. A. van der Spek<\/strong>, Problem 1018 and Solution, <em>Crux Mathematicorum<\/em> 12 (1986) 176-179.<\/p><\/blockquote>\n<p>An accomplished amateur geometer, Schiffler discovered one of the most attractive of the &#8220;twentieth-century&#8221; triangle centers, now known as the <em>Schiffler Point<\/em>.<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/cms.ferzkopp.net\/images\/Schiffler_Point.png\" alt=\"Schiffler Point\" \/><\/p>\n<p>Let I denote the incenter of a triangle ABC. The Schiffler point of ABC is the point of concurrence of the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Euler_line\">Euler lines<\/a> of the four triangles BCI, CAI, ABI, ABC. Trilinear coordinates for the Schiffler point are<\/p>\n<blockquote><p>1\/(cos(B) + cos(C)) :<br \/>\n1\/(cos(C) + cos(A)) :<br \/>\n1\/(cos(A) + cos(B)),<\/p><\/blockquote>\n<p>or, equivalently,<\/p>\n<blockquote><p>(b + c &#8211; a)\/(b + c) :<br \/>\n(c + a &#8211; b)\/(c + a) :<br \/>\n(a + b &#8211; c)\/(a + b),<\/p><\/blockquote>\n<p>where a, b, c denote the sidelengths of triangle ABC.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A Schiffler Point? That is cool!&#8230; Any relationships to persons of the same name (like myself) are purely coincidental.<\/p>\n","protected":false},"author":1,"featured_media":155,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[2],"tags":[18,68],"class_list":["post-28","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-site","tag-geometry","tag-mathematics"],"_links":{"self":[{"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/posts\/28","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/comments?post=28"}],"version-history":[{"count":3,"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/posts\/28\/revisions"}],"predecessor-version":[{"id":156,"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/posts\/28\/revisions\/156"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/media\/155"}],"wp:attachment":[{"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/media?parent=28"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/categories?post=28"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ferzkopp.net\/wordpress\/wp-json\/wp\/v2\/tags?post=28"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}