The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Triangle Proof: r + r1 + r2 = CD

Date: 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

                                  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 

     r+r1+r2 = (1 + AC/AB + BC/AB)*r

               AB + AC + BC      AC*BC
             = ------------ * ------------
                    AB        AB + AC + BC

             = -----

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   
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.