{"id":758,"date":"2019-03-20T10:43:37","date_gmt":"2019-03-20T14:43:37","guid":{"rendered":"http:\/\/josmfs.net\/?p=758"},"modified":"2019-05-11T12:54:04","modified_gmt":"2019-05-11T16:54:04","slug":"straight-and-narrow-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/03\/20\/straight-and-narrow-problem\/","title":{"rendered":"Straight and Narrow Problem"},"content":{"rendered":"<p><a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/03\/Straight-and-Narrow.gif\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-902\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/04\/Straight-Narrow-Fig3.jpg\" alt=\"\" width=\"200\" height=\"161\" \/><\/a>The following interesting behavior was found at the <a href=\"http:\/\/www.futilitycloset.com\/2016\/01\/11\/straight-and-narrow-4\/\">Futility Closet<\/a> website:<\/p>\n<p>\u201cA pleasing fact from <a href=\"http:\/\/amendavidwells.blogspot.com\/\">David Wells\u2019 Archimedes Mathematics Education Newsletter<\/a>: Draw two parallel lines. Fix a point A on one line and move a second point B along the other line. If an equilateral triangle is constructed with these two points as two of its vertices, then as the second point moves, the third vertex C of the triangle will trace out a straight line. Thanks to reader Matthew Scroggs for the tip and the GIF.\u201d<\/p>\n<p>This is rather amazing and cries out for a proof. It also raises the question of how anyone noticed this behavior in the first place. I proved the result with calculus, but I wonder if there is a slicker way that makes it more obvious. See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/03\/Straight-and-Narrow-160205.pdf\">Straight and Narrow Problem<\/a>.<\/p>\n<p><strong>(Update 3\/25\/2019)<\/strong><!--more-->In discovering that the David Wells of the \u201cRegiomontanus 1471 Problem\u201d was the same person referenced in the \u201cStraight and Narrow\u201d Futility Closet article, I was able to find the exact reference Futility Closet used, namely,<\/p>\n<p>Wells, David, \u201cAn equilateral triangle on three parallel lines,\u201d Archimedes Mathematics Education Newsletter, #1 January 04 2016, p.35 (https:\/\/drive.google.com\/file\/d\/0BwT4tltuHZVdME0weEJLWWdlVGM\/view, retrieved 3\/25\/2019)<\/p>\n<p>In his article Wells provides the geometric argument I was wondering about. Rather than reproduce the discussion, I recommend referring to the original article, which has a much broader exploration of the matter. Ironically, I am reminded of the situation in Newton\u2019s Principia where he uses plane geometric arguments in place of the calculus he so recently espoused. It may be a matter of taste which is preferable: geometry or calculus. I have to admit in both cases, Newton\u2019s and the \u201cStraight and Narrow Problem\u201d, I prefer the calculus as simpler and more direct.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The following interesting behavior was found at the Futility Closet website: \u201cA pleasing fact from David Wells\u2019 Archimedes Mathematics Education Newsletter: Draw two parallel lines. Fix a point A on one line and move a second point B along the other line. If an equilateral triangle is constructed with these two points as two of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[32,115,16,93,13],"class_list":["post-758","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-calculus","tag-david-wells","tag-futility-closet","tag-matthew-scroggs","tag-plane-geometry"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/758","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=758"}],"version-history":[{"count":9,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/758\/revisions"}],"predecessor-version":[{"id":986,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/758\/revisions\/986"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=758"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=758"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=758"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}