{"id":469,"date":"2019-01-11T11:09:54","date_gmt":"2019-01-11T16:09:54","guid":{"rendered":"http:\/\/josmfs.net\/?p=469"},"modified":"2021-06-12T15:06:27","modified_gmt":"2021-06-12T19:06:27","slug":"nahin-triangle-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/01\/11\/nahin-triangle-problem\/","title":{"rendered":"Nahin Triangle Problem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-2241\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/06\/Nahin-Triangle-Fig2.jpg\" alt=\"\" width=\"202\" height=\"173\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/06\/Nahin-Triangle-Fig2.jpg 400w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/06\/Nahin-Triangle-Fig2-300x257.jpg 300w\" sizes=\"auto, (max-width: 202px) 100vw, 202px\" \/>This article is basically a technical footnote without wider significance. At the time I had been reading with interest Paul J. Nahin\u2019s latest book <em>Number-Crunching<\/em> (2011). Nahin presents a problem that he will solve with the Monte Carlo sampling approach.<\/p>\n<p>\u201cTo start, imagine an equilateral triangle with side lengths 2. If we pick a point \u2018at random\u2019 from the interior of the triangle, what is the probability that the point is no more distant than d = \u221a2 from each of the triangle\u2019s three vertices? The shaded region in the figure is where all such points are located.\u201d<\/p>\n<p>Nahin provided a theoretical calculation for the answer and said that it \u201crequires mostly only high school geometry, plus one step that I think requires a simple freshman calculus computation.\u201d This article presents my solution without calculus. See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/01\/Nahin-Triangle-Problem-120229.pdf\">Nahin Triangle Problem<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This article is basically a technical footnote without wider significance. At the time I had been reading with interest Paul J. Nahin\u2019s latest book Number-Crunching (2011). Nahin presents a problem that he will solve with the Monte Carlo sampling approach. \u201cTo start, imagine an equilateral triangle with side lengths 2. If we pick a point [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[69,13],"class_list":["post-469","post","type-post","status-publish","format-standard","hentry","category-curiosities-and-questions","tag-paul-nahin","tag-plane-geometry"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/469","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=469"}],"version-history":[{"count":5,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/469\/revisions"}],"predecessor-version":[{"id":2242,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/469\/revisions\/2242"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=469"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=469"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=469"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}