{"id":1427,"date":"2020-01-16T09:53:42","date_gmt":"2020-01-16T14:53:42","guid":{"rendered":"http:\/\/josmfs.net\/?p=1427"},"modified":"2021-05-17T20:31:27","modified_gmt":"2021-05-18T00:31:27","slug":"amazing-triangle-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2020\/01\/16\/amazing-triangle-problem\/","title":{"rendered":"Amazing Triangle Problem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-2139\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/05\/Amazing-Triangle-Fig2.jpg\" alt=\"\" width=\"249\" height=\"250\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/05\/Amazing-Triangle-Fig2.jpg 500w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/05\/Amazing-Triangle-Fig2-298x300.jpg 298w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/05\/Amazing-Triangle-Fig2-150x150.jpg 150w\" sizes=\"auto, (max-width: 249px) 100vw, 249px\" \/>Here is another simply amazing problem from <em>Five Hundred Mathematical Challenges<\/em>:<\/p>\n<p>\u201c<strong>Problem 154<\/strong>. Show that three solutions, (x1,<span style=\"color: #ffffff;\">.<\/span>y1), (x2,<span style=\"color: #ffffff;\">.<\/span>y2), (x3, y3), of the four solutions of the simultaneous equations<br \/><span style=\"color: #ffffff;\">____________<\/span>(x \u2013 h)\u00b2 + (y \u2013 k)\u00b2 = 4(h\u00b2 + k\u00b2) <br \/><span style=\"color: #ffffff;\">______________________<\/span>xy = hk<br \/>are vertices of an equilateral triangle. Give a geometrical interpretation.\u201d<\/p>\n<p>Again, I don\u2019t see how anyone could have discovered this property involving a circle, a hyperbola, and an equilateral triangle. It seems plausible when h<span style=\"color: #ffffff;\">.<\/span>=<span style=\"color: #ffffff;\">.<\/span>k, but it is not at all obvious for h<span style=\"color: #ffffff;\">.<\/span>\u2260<span style=\"color: #ffffff;\">.<\/span>k. For some reason, I had difficulty getting a start on a solution, until the obvious approach dawned on me. I don\u2019t know why it took me so long.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/01\/Amazing-Triangle-Problem-191212.pdf\">Amazing Triangle Problem<\/a>.<\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is another simply amazing problem from Five Hundred Mathematical Challenges: \u201cProblem 154. Show that three solutions, (x1,.y1), (x2,.y2), (x3, y3), of the four solutions of the simultaneous equations____________(x \u2013 h)\u00b2 + (y \u2013 k)\u00b2 = 4(h\u00b2 + k\u00b2) ______________________xy = hkare vertices of an equilateral triangle. Give a geometrical interpretation.\u201d Again, I don\u2019t see [&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":[177,40],"class_list":["post-1427","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-500-math-challenges","tag-analytic-geometry"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1427","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=1427"}],"version-history":[{"count":4,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1427\/revisions"}],"predecessor-version":[{"id":2140,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1427\/revisions\/2140"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=1427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=1427"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=1427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}