{"id":1291,"date":"2019-10-14T15:15:48","date_gmt":"2019-10-14T19:15:48","guid":{"rendered":"http:\/\/josmfs.net\/?p=1291"},"modified":"2021-09-27T18:32:56","modified_gmt":"2021-09-27T22:32:56","slug":"mysterious-doppelganger-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/10\/14\/mysterious-doppelganger-problem\/","title":{"rendered":"Mysterious Doppelg\u00e4nger Problem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-1392\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/12\/Mysterious-Doppleganger-Fig-rev.jpg\" alt=\"\" width=\"250\" height=\"254\">I found this problem from the Math Challenges section of the 2002 <em>Pi in the Sky<\/em> Canadian math magazine for high school students to be truly astonishing.<\/p>\n<p>\u201c<strong>Problem 4<\/strong>. Inside of the square ABCD, take any point P. Prove that the perpendiculars from A on BP, from B on CP, from C on DP, and from D on AP are concurrent (i.e. they meet at one point).\u201d<\/p>\n<p>How could such a complicated arrangement produce such an amazing result? I didn\u2019t know where to begin to try to prove it. My wandering path to discovery produced one of my most satisfying \u201caha!\u201d moments.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/09\/Mysterious-Doppelganger-Problem-191227rev.pdf\">Mysterious Doppelg\u00e4nger Problem<\/a><\/p>\n<p><strong>Update (12\/27\/2019)<\/strong> I goofed.&nbsp; I had plotted the original figure incorrectly. (No figure was given in the <em>Pi in the Sky <\/em>statement of the problem.) Fortunately, the original solution idea still worked.<\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I found this problem from the Math Challenges section of the 2002 Pi in the Sky Canadian math magazine for high school students to be truly astonishing. \u201cProblem 4. Inside of the square ABCD, take any point P. Prove that the perpendiculars from A on BP, from B on CP, from C on DP, and [&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":[162,13],"class_list":["post-1291","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-pi-in-the-sky","tag-plane-geometry"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1291","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=1291"}],"version-history":[{"count":8,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1291\/revisions"}],"predecessor-version":[{"id":2403,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1291\/revisions\/2403"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=1291"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=1291"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=1291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}