{"id":2896,"date":"2022-10-01T08:42:38","date_gmt":"2022-10-01T12:42:38","guid":{"rendered":"https:\/\/josmfs.net\/?p=2896"},"modified":"2022-10-04T07:25:28","modified_gmt":"2022-10-04T11:25:28","slug":"neubergs-theorem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2022\/10\/01\/neubergs-theorem\/","title":{"rendered":"Neuberg\u2019s Theorem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-2894\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2022\/10\/Neubergs-Theorem-Fig2-854x1024.jpg\" alt=\"\" width=\"249\" height=\"298\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2022\/10\/Neubergs-Theorem-Fig2-854x1024.jpg 854w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2022\/10\/Neubergs-Theorem-Fig2-250x300.jpg 250w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2022\/10\/Neubergs-Theorem-Fig2-768x921.jpg 768w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2022\/10\/Neubergs-Theorem-Fig2-624x748.jpg 624w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2022\/10\/Neubergs-Theorem-Fig2.jpg 1004w\" sizes=\"auto, (max-width: 249px) 100vw, 249px\" \/>This turned out to be a challenging puzzle from the 1980 Canadian Math Society\u2019s magazine, <em>Crux Mathematicorum<\/em>.<\/p>\n<p>\u201c<em>Proposed by Leon Bankoff, Los Angeles<\/em>, <em>California.<\/em><\/p>\n<p>Professor Euclide Paracelso Bombasto Umbugio has once again retired to his <em>tour d\u2019ivoire <\/em>where he is now delving into the supersophisticated intricacies of the works of Grassmann, as elucidated by Forder\u2019s <em>Calculus of Extension. <\/em>His goal is to prove Neuberg\u2019s Theorem:<\/p>\n<p><em>If <\/em>D, E, F <em>are the centers of squares described externally on the sides of a triangle <\/em>ABC, <em>then the midpoints of these sides are the centers of squares described internally on the sides of triangle <\/em>DEF.\u00a0 [The accompanying diagram shows only one internally described square.]<\/p>\n<p>Help the dedicated professor emerge from his self-imposed confinement and enjoy the thrill of hyperventilation by showing how to solve his problem using only highschool, synthetic, Euclidean, \u2018plain\u2019 geometry.\u201d<\/p>\n<p>Alas, my plane geometry capability was inadequate to solve the puzzle that way, so I had to resort to the sledge hammer of analytic geometry, trigonometry, and complex variables.<\/p>\n<p>See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2022\/10\/Neubergs-Theorem-220925rev.pdf\">Neuberg\u2019s Theorem<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This turned out to be a challenging puzzle from the 1980 Canadian Math Society\u2019s magazine, Crux Mathematicorum. \u201cProposed by Leon Bankoff, Los Angeles, California. Professor Euclide Paracelso Bombasto Umbugio has once again retired to his tour d\u2019ivoire where he is now delving into the supersophisticated intricacies of the works of Grassmann, as elucidated by Forder\u2019s [&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":[51,212,13,41],"class_list":["post-2896","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-complex-numbers","tag-crux-mathematicorum","tag-plane-geometry","tag-trigonometry"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2896","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=2896"}],"version-history":[{"count":3,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2896\/revisions"}],"predecessor-version":[{"id":2903,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2896\/revisions\/2903"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=2896"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=2896"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=2896"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}