{"id":1264,"date":"2019-09-18T18:24:50","date_gmt":"2019-09-18T22:24:50","guid":{"rendered":"http:\/\/josmfs.net\/?p=1264"},"modified":"2019-09-18T18:24:50","modified_gmt":"2019-09-18T22:24:50","slug":"circular-rendezvous-mystery","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/09\/18\/circular-rendezvous-mystery\/","title":{"rendered":"Circular Rendezvous Mystery"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-1262\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/09\/Circular-Rendezvous-Fig.jpg\" alt=\"\" width=\"200\" height=\"252\" \/>Here is yet another surprising result from Colin Hughes at <em>Maths Challenge<\/em>.<\/p>\n<p>\u201c<strong>Problem<\/strong><br \/>\nIt can be shown that a unique circle passes through three given points. In triangle ABC three points A\u2019, B\u2019, and C\u2019 lie on the edges opposite A, B, and C respectively. Given that the circle AB\u2019C\u2019 intersects circle BA\u2019C\u2019 inside the triangle at point P, prove that circle CA\u2019B\u2019 will be concurrent with P.\u201d<\/p>\n<p>I have to admit it took me a while to arrive at the final version of my proof. My original approach had some complicated expressions using various angles, and then I realized I had not used one of my assumptions. Once I did, all the complications faded away and the result became clear.<\/p>\n<p>See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/09\/Circular-Rendezvous-Mystery-190825.pdf\">Circular Rendezvous Mystery<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is yet another surprising result from Colin Hughes at Maths Challenge. \u201cProblem It can be shown that a unique circle passes through three given points. In triangle ABC three points A\u2019, B\u2019, and C\u2019 lie on the edges opposite A, B, and C respectively. Given that the circle AB\u2019C\u2019 intersects circle BA\u2019C\u2019 inside the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[151,13],"class_list":["post-1264","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-colin-hughes","tag-plane-geometry"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1264","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=1264"}],"version-history":[{"count":1,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1264\/revisions"}],"predecessor-version":[{"id":1265,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1264\/revisions\/1265"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=1264"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=1264"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=1264"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}