tag:blogger.com,1999:blog-6933544261975483399.post1309513501147655717..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Problem 28. Right Triangle, altitude, incircles, inradiusAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-6933544261975483399.post-57809427743297033362019-08-06T05:35:36.524-07:002019-08-06T05:35:36.524-07:00How to find the inradius of tr. ABC?
How to find the inradius of tr. ABC?<br /> Rniehttps://www.blogger.com/profile/15012247681944808200noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-20279865674045382272018-01-09T08:57:45.299-08:002018-01-09T08:57:45.299-08:00AB=a , BC =b , AD= n , DC= m and BD = h
For the ...AB=a , BC =b , AD= n , DC= m and BD = h <br /><br />For the radius of the circles we have the following equations :<br /><br />2r=a+b-m-n<br />2r1 = n+h-a<br />2r2 =m+h-b<br /><br />DC= m and EC= b-r so we get :<br /><br />DE =DC-EC=m-(b-r)<br /><br />using the formula for 2r :<br /><br />2DE=2m-2b+2r<br />2DE=2m-2b+a+b-n-m<br />2DE=m-b+a-n<br />2DE=m+h-b+a-h-n<br />2DE=m+h-b-(n+h-a)<br /><br /><br />m+h-b=2r2 and n+h-a =2r1 therefore we get :<br /><br />2DE=2r2-2r1<br />DE=r2-r1<br />Arifhttps://www.blogger.com/profile/16774426807135540272noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-61925949270261514392009-05-29T22:12:58.617-07:002009-05-29T22:12:58.617-07:00Let incircle of Tr. ABC meet AB in H. Thus, AE = A...Let incircle of Tr. ABC meet AB in H. Thus, AE = AH = AB - BH = c - r = c - (c+a-b)/2 = (c-a+b)/2.<br />From Tr ABD, AD = c^2/b. Hence, DE = AE - AD = (c-a+b)/2 -c^2/b = (bc -ab+b^2-2c^2)/2 ------(1)<br />Triangles ABD & CBD give 2r2=CD+BD -a and 2r1= AD+BD -c whence 2r2 -2r1 = CD - AD + c - a = a^2/b - c^2/b + c -a or r2-r1=(bc-ab+a^2-c^2)/2b = (bc-ab+b^2-2c^2)/2---(2) since b^2=a^2+c^2. Equations (1) & (2) give us r2 -r1 = DE whereas it is appaent from the diagram that FG =r2 - r1. Hence, DE = FG = r2 - r1<br />Vihaan: vihaanup@gmail.comVihaanhttps://www.blogger.com/profile/14122321143712979648noreply@blogger.com