Associated Topics || Dr. Math Home || Search Dr. Math

### Triangle Proof: r + r1 + r2 = CD

```
Date: 04/20/2001 at 11:15:37
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/
```
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
Math Forum Home || Math Library || Quick Reference || Math Forum Search