{"id":2899,"date":"2022-10-07T00:06:00","date_gmt":"2022-10-07T04:06:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2899"},"modified":"2024-10-25T10:57:16","modified_gmt":"2024-10-25T14:57:16","slug":"distance-between-the-incenter-and-circumcenter","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2899","title":{"rendered":"Distance between the incenter and circumcenter"},"content":{"rendered":"\n<p>The distance between the incenter and circumcenter of a triangle, is calculated by <a href=\"https:\/\/en.wikipedia.org\/wiki\/Euler%27s_theorem_in_geometry\">Euler&#8217;s theorem in geometry<\/a>: $$d=\\sqrt{R(R-2r)}$$ which also implies $$R\\ge2r$$<\/p>\n\n\n\n<p>For a <a href=\"http:\/\/Bicentric quadrilateral\">Bicentric quadrilateral<\/a>, the distance between the incenter and circumcenter can be calculated by <a href=\"https:\/\/en.wikipedia.org\/wiki\/Bicentric_quadrilateral#Fuss'_theorem\">Fuss&#8217;s theorem<\/a> or <a href=\"https:\/\/en.wikipedia.org\/wiki\/Bicentric_quadrilateral#Carlitz'_identity\">Carlitz&#8217; identity<\/a>.<\/p>\n\n\n\n<p>Using Euler&#8217;s theorem in geometry, it can be approved that for $\\triangle{ABC}$ $$1\\le\\cos{A}+\\cos{B}+\\cos{C}\\le\\dfrac{3}{2}$$ $$0\\le\\sin{\\dfrac{A}{2}}\\cdot\\sin{\\dfrac{B}{2}}\\cdot\\sin{\\dfrac{C}{2}}\\le\\dfrac{1}{8}$$Because $$\\cos{A}+\\cos{B}+\\cos{C}=1+\\dfrac{r}{R}$$ $$4\\cdot\\sin{\\dfrac{A}{2}}\\cdot\\sin{\\dfrac{B}{2}}\\cdot\\sin{\\dfrac{C}{2}}=\\dfrac{r}{R}$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The distance between the incenter and circumcenter of a triangle, is calculated by Euler&#8217;s theorem in geometry: $$d=\\sqrt{R(R-2r)}$$ which also implies $$R\\ge2r$$ For a Bicentric quadrilateral, the distance between the incenter and circumcenter can be calculated by Fuss&#8217;s theorem or &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=2899\">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":[9,15],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2899"}],"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=2899"}],"version-history":[{"count":13,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2899\/revisions"}],"predecessor-version":[{"id":2912,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2899\/revisions\/2912"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2899"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2899"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2899"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}