Triangle Proof: r + r1 + r2 = CDDate: 04/20/2001 at 11:15:37 From: Rebecca Prochaska Subject: Geometry How do you prove the following: Let CD be an altitude of triangle ABC, and assume that angle C is 90 degrees. Let r1 and r2 be the inradii of triangle CAD and triangle CBD, respectively. Show that r+r1+r2 = CD, where r is the inradius of triangle ABC. Date: 04/20/2001 at 14:13:50 From: Doctor Floor Subject: Re: Geometry Hi, Rebecca, Thanks for writing. Let's consider the incenter I of triangle ABC. We can divide ABC into three triangles IBC, AIC and ABI. From the fact that ABC has the same area as these three taken together, and from the fact that r is an altitude in all three of them, we can conclude that: 0.5*r*AB + 0.5*r*BC + 0.5*r*AC = 0.5AC*BC r*(AB + AC + BC) = AC*BC AC*BC r = ------------ AB + AC + BC We know that triangles CAD and CBD are similar to ABC and that their ratios of similarity are AC:AB and BC:AB, respectively. From this we find: r+r1+r2 = (1 + AC/AB + BC/AB)*r AB + AC + BC AC*BC = ------------ * ------------ AB AB + AC + BC AC*BC = ----- AB Since the area of ABC is 0.5*AC*BC as well as 0.5*AB*CD, the latter is equal to CD, as desired. If you need more help, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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