{"id":4839,"date":"2025-01-12T18:44:00","date_gmt":"2025-01-12T22:44:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=4839"},"modified":"2025-02-24T18:46:56","modified_gmt":"2025-02-24T22:46:56","slug":"geometry-challenge-17","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=4839","title":{"rendered":"Geometry Challenge &#8211; 17"},"content":{"rendered":"\n<p>Let $D$ be an interior point inside equilateral $\\triangle{ABC}$, so that $\\angle{BDC}=150^\\circ$. Prove that the line segment $AD$, $BD$ and $CD$ are the sides of a right triangle. Click <a href=\"javascript:toggle_visibility('geo-chall-17-proof')\">here<\/a> for the proof.<\/p>\n\n\n\n<div id=\"geo-chall-17-proof\" style=\"display:none\">\n\n\n\n<p><strong>Proof:<\/strong> Rotating $\\triangle{ADC}$ counter-clock-wise around $C$ by $60^\\circ$, so that $D$ is moved to $D&#8217;$, and $A$ to $B$, we have: <\/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\/2025\/02\/Screenshot-2025-02-24-at-3.47.13\u202fPM-1024x942.png\" alt=\"\" class=\"wp-image-4840\" width=\"256\" height=\"236\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-24-at-3.47.13\u202fPM-1024x942.png 1024w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-24-at-3.47.13\u202fPM-300x276.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-24-at-3.47.13\u202fPM-768x707.png 768w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-24-at-3.47.13\u202fPM.png 1256w\" sizes=\"(max-width: 256px) 100vw, 256px\" \/><\/figure>\n\n\n\n<p>Because $\\angle{DCD&#8217;}=60^\\circ$, and $CD=CD&#8217;$, $\\triangle{DCD&#8217;}$ is equilateral. Therefore, $CD=DD&#8217;$, and $\\angle{CDD&#8217;}=60^\\circ$.<\/p>\n\n\n\n<p>Because $AD=BD&#8217;$, therefore the line segment $AD$, $BD$ and $CD$ forms $\\triangle{DBD&#8217;}$.<\/p>\n\n\n\n<p>Because $\\angle{BDC}=150^\\circ$, $\\angle{CDD&#8217;}=60^\\circ$, we have $$\\angle{BDD&#8217;}=\\angle{BDC}-\\angle{CDD&#8217;}=150^\\circ-60^\\circ=90^\\circ$$<\/p>\n\n\n\n<p>Therefore, $\\triangle{DBD&#8217;}$ is right, implying the line segment $AD$, $BD$ and $CD$ are the sides of a right triangle.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Let $D$ be an interior point inside equilateral $\\triangle{ABC}$, so that $\\angle{BDC}=150^\\circ$. Prove that the line segment $AD$, $BD$ and $CD$ are the sides of a right triangle. Click here for the proof. Proof: Rotating $\\triangle{ADC}$ counter-clock-wise around $C$ by &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=4839\">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\/4839"}],"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=4839"}],"version-history":[{"count":6,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4839\/revisions"}],"predecessor-version":[{"id":4846,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4839\/revisions\/4846"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4839"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4839"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4839"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}