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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

                                         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

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