{"id":650,"date":"2013-07-27T14:20:54","date_gmt":"2013-07-27T19:20:54","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=650"},"modified":"2013-07-28T15:22:11","modified_gmt":"2013-07-28T20:22:11","slug":"pww-nine-point-circle-property","status":"publish","type":"post","link":"http:\/\/daylateanddollarshort.com\/bloog\/pww-nine-point-circle-property\/","title":{"rendered":"Proof Without Words: Nine-Point Circle Property"},"content":{"rendered":"<p>The <a title=\"&quot;Nine-point circle&quot; at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Nine-point_circle\">&#8220;Nine-Point Circle&#8221;<\/a> of a triangle \\(\\triangle ABC\\) contains the vertices of the medial triangle (\\(\\triangle RST \\)) and orthic triangle (\\(\\triangle UVW \\)), as well as points \\(X\\), \\(Y\\), \\(Z\\) that bisect segments from \\(\\triangle ABC\\)&#8217;s orthocenter (\\(H\\)) to its vertices.<\/p>\n<p><a title=\"Nine-point circle question on Math.StackExchange.com\" href=\"http:\/\/math.stackexchange.com\/q\/451979\/409\">A question on Math.StackExchange.com<\/a>\u00a0asked for proof that the center of the nine-point circle bisects the segment joining orthocenter \\(H\\) and circumcenter \\(O\\).\u00a0I thought I&#8217;d post my answer here.<\/p>\n<p><a href=\"http:\/\/daylateanddollarshort.com\/bloog\/wp-content\/uploads\/2013\/07\/bloog-nineptcircle1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-661 aligncenter\" alt=\"bloog-nineptcircle\" src=\"http:\/\/daylateanddollarshort.com\/bloog\/wp-content\/uploads\/2013\/07\/bloog-nineptcircle1-300x120.png\" width=\"450\" height=\"180\" srcset=\"http:\/\/daylateanddollarshort.com\/bloog\/wp-content\/uploads\/2013\/07\/bloog-nineptcircle1-300x120.png 300w, http:\/\/daylateanddollarshort.com\/bloog\/wp-content\/uploads\/2013\/07\/bloog-nineptcircle1.png 600w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The &#8220;Nine-Point Circle&#8221; of a triangle \\(\\triangle ABC\\) contains the vertices of the medial triangle (\\(\\triangle RST \\)) and orthic triangle (\\(\\triangle UVW \\)), as well as points \\(X\\), \\(Y\\), \\(Z\\) that bisect segments from \\(\\triangle ABC\\)&#8217;s orthocenter (\\(H\\)) to its vertices. A question on Math.StackExchange.com\u00a0asked for proof that the center of the nine-point circle [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7,6,14,5],"tags":[],"class_list":["post-650","post","type-post","status-publish","format-standard","hentry","category-classroom","category-misc-math","category-proof-without-words","category-trigonography"],"_links":{"self":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/650","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/comments?post=650"}],"version-history":[{"count":10,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/650\/revisions"}],"predecessor-version":[{"id":667,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/650\/revisions\/667"}],"wp:attachment":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/media?parent=650"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/categories?post=650"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/tags?post=650"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}