{"id":2562,"date":"2022-03-21T19:52:00","date_gmt":"2022-03-21T23:52:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2562"},"modified":"2024-11-21T14:52:31","modified_gmt":"2024-11-21T18:52:31","slug":"geometry-challenge-11","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2562","title":{"rendered":"Geometry Challenge &#8211; 11"},"content":{"rendered":"\n<p>Let $D$ be an arbitrary point on the side $BC$ of a given triangle $ABC$ and let $E$ be the intersection of $AD$ and the second external common tangent of the incircles of triangles $ABD$ and ACD. As $D$ assumes all positions between $B$ and $C$, prove that the point $E$ traces an arc of a circle. Click <a href=\"javascript:toggle_visibility('geo-chall-11-problem');toggle_visibility('geo-chall-11-solution');\">here<\/a> for the proof.<\/p>\n\n\n\n<div id=\"geo-chall-11-problem\" name=\"geo-chall-11-problem\" style=\"display:block\">\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" src=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.18.05-PM-1024x732.png\" alt=\"\" class=\"wp-image-2563\" width=\"256\" height=\"183\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.18.05-PM-1024x732.png 1024w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.18.05-PM-300x214.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.18.05-PM-768x549.png 768w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.18.05-PM.png 1080w\" sizes=\"(max-width: 256px) 100vw, 256px\" \/><\/figure>\n\n\n\n<\/div>\n<div id=\"geo-chall-11-solution\" name=\"geo-chall-11-solution\" style=\"display:none\">\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" src=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.06.10-PM-1024x732.png\" alt=\"\" class=\"wp-image-2565\" width=\"256\" height=\"183\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.06.10-PM-1024x732.png 1024w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.06.10-PM-300x214.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.06.10-PM-768x549.png 768w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-5.06.10-PM.png 1080w\" sizes=\"(max-width: 256px) 100vw, 256px\" \/><\/figure>\n\n\n\n<p><strong>Proof:<\/strong> Mark various tangent points as the above. We have $JK=ML$. $$\\because JK=KE+JE=EN+EO=EN+EN+NO=2EN+NO$$ and $$ ML=LD+MD=DO+DN=DO+DO+NO=2DO+NO$$ Therefore $EN=DO$, $KE=DL$, and $JE=MD$.<\/p>\n\n\n\n<p>$$\\because AE=AN-EN=AG-KE=AC-CG-DL=AC-CM-DL$$ and$$AE=AO-EO=AF-JE=AB-BF-MD=AB-BL-MD$$<\/p>\n\n\n\n<p>We have $$2AE=AC-CM-DL+AB-BL-MD$$ $$=AB+AC-(CM+MD+DL+BL)=AB+AC-BC$$ Therefore $$AE=\\dfrac{AB+AC-BC}{2}$$<\/p>\n\n\n\n<p>which implies that the length of $AE$ is a constant for $\\triangle{ABC}$, and $E$ is on an arc of a circle, as shown below, where $P$ and $Q$ are tangent points of the in-circle of $\\triangle{ABC}$ on $AC$ and $AB$ respectively.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" src=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-7.36.50-PM-1024x736.png\" alt=\"\" class=\"wp-image-2570\" width=\"256\" height=\"184\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-7.36.50-PM-1024x736.png 1024w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-7.36.50-PM-300x216.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-7.36.50-PM-768x552.png 768w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2022\/04\/Screen-Shot-2022-04-10-at-7.36.50-PM.png 1080w\" sizes=\"(max-width: 256px) 100vw, 256px\" \/><\/figure>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Let $D$ be an arbitrary point on the side $BC$ of a given triangle $ABC$ and let $E$ be the intersection of $AD$ and the second external common tangent of the incircles of triangles $ABD$ and ACD. As $D$ assumes &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=2562\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[9],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2562"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2562"}],"version-history":[{"count":15,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2562\/revisions"}],"predecessor-version":[{"id":4711,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2562\/revisions\/4711"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2562"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2562"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2562"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}