{"id":887,"date":"2019-04-05T19:22:50","date_gmt":"2019-04-05T23:22:50","guid":{"rendered":"http:\/\/josmfs.net\/?p=887"},"modified":"2020-05-15T10:32:55","modified_gmt":"2020-05-15T14:32:55","slug":"magic-parallelogram","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/04\/05\/magic-parallelogram\/","title":{"rendered":"Magic Parallelogram"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-885\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/04\/Magic-Parallelogram-Fig.jpg\" alt=\"\" width=\"200\" height=\"110\" \/>I came across this problem in Alfred Posamentier\u2019s book, but I remember I had seen it a couple of places before and had never thought to solve it. At first, it seems like magic.<\/p>\n<p>In <em>any<\/em> convex quadrilateral (line between any two points in the quadrilateral lies entirely inside the quadrilateral) inscribe a second convex quadrilateral with its vertices on the midpoints of the sides of the first quadrilateral. Show that the inscribed quadrilateral must be a parallelogram.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/04\/Magic-Parallelogram-190405.pdf\">Magic Parallelogram<\/a>.<\/p>\n<p><strong>(Update 5\/15\/2020)<\/strong> <!--more--><a href=\"http:\/\/mscroggs.co.uk\/\">Matthew Scroggs<\/a> has a <a href=\"https:\/\/aperiodical.com\/2020\/05\/the-big-lock-down-math-off-match-18\/\">different proof<\/a> of the Magic Parallelogram for <a href=\"https:\/\/aperiodical.com\/2020\/05\/the-big-lock-down-math-off-match-18\/\">The Aperiodical<\/a>\u2019s Big Math Lockdown contest.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I came across this problem in Alfred Posamentier\u2019s book, but I remember I had seen it a couple of places before and had never thought to solve it. At first, it seems like magic. In any convex quadrilateral (line between any two points in the quadrilateral lies entirely inside the quadrilateral) inscribe a second convex [&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":[132,13,31],"class_list":["post-887","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-alfred-posamentier","tag-plane-geometry","tag-vectors"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/887","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=887"}],"version-history":[{"count":5,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/887\/revisions"}],"predecessor-version":[{"id":1608,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/887\/revisions\/1608"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=887"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=887"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=887"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}