{"id":229,"date":"2019-02-08T14:57:12","date_gmt":"2019-02-08T20:57:12","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?p=229"},"modified":"2019-02-22T14:12:18","modified_gmt":"2019-02-22T20:12:18","slug":"problem-12-february-8th-2019","status":"publish","type":"post","link":"https:\/\/blogs.acu.edu\/mu_sigma\/2019\/02\/08\/problem-12-february-8th-2019\/","title":{"rendered":"Problem 12 – February 8th, 2019 – Is It Really 1\/7?"},"content":{"rendered":"

Problem 12 \u2013 February 8th, 2019<\/b><\/p>\n

Is It Really One-seventh?<\/b><\/span><\/p>\n

I learned of this problem in a session on mathematics and art at the 2003 joint AMS\/MAA meeting.<\/i><\/p>\n

Given any triangle ABC<\/i>, construct a line from each vertex to the opposite side as indicated in the figure below. If point D<\/i> is one-third of the way from point B<\/i> to C<\/i>, point E<\/i> is one-third of the way from C<\/i> to A<\/i>, and point F<\/i> is one-third of the way from A<\/i> to B<\/i>, show that the triangle GHI<\/i> has area one-seventh of the area of triangle ABC<\/i>.<\/p>\n

<\/p>\n

Submit your answers to mathpotw@acu.edu. \u00a0Details for submissions can be found\u00a0here<\/a>.<\/p>\n

Correct solutions were submitted by Yunxi Wei.<\/p>\n

Solution to Is It Really One-seventh?<\/b><\/span><\/p>\n

We get a solution by tiling the plane using copies of the triangle GHI<\/i>. The solution becomes apparent by looking at three parallelograms in the figure below, each of which has one of the sides of the triangle ABC<\/i> as a diagonal.<\/p>\n

In particular, look at the parallelogram BJCH<\/i>. The area of the parallelogram is four times the area of triangle GHI<\/i>, and by the apparent symmetry, the part of the parallelogram inside triangle ABC<\/i> has area two times the area of triangle GHI<\/i>. Similar arguments hold for parallelogram AKBG<\/i> and parallelogram AICL<\/i>.<\/p>\n

Thus the area inside of triangle ABC<\/i> is seven times the area of triangle GHI<\/i>, and the problem is solved.<\/p>\n


\nHmmm! I wonder what would happen if we went 1\/4 of the way along each side? Or how about 1\/n th of the way?<\/p>\n","protected":false},"excerpt":{"rendered":"

Problem 12 \u2013 February 8th, 2019 Is It Really One-seventh? I learned of this problem in a session on mathematics and art at the 2003 joint AMS\/MAA meeting. Given any triangle ABC, construct a line from each vertex to the opposite side as indicated in the figure below. If point D is one-third of the […]<\/p>\n","protected":false},"author":130,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[181534],"tags":[],"class_list":["post-229","post","type-post","status-publish","format-standard","hentry","category-problem-of-the-week"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/229","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/users\/130"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/comments?post=229"}],"version-history":[{"count":6,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/229\/revisions"}],"predecessor-version":[{"id":238,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/229\/revisions\/238"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=229"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/categories?post=229"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/tags?post=229"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}