Find the Radius of the Circles

Date: 11/03/2002 at 12:11:21
From: Dan
Subject: Geometry

Given: a 3,4,5 triangle, and inside it, inscribed, two circles of 
equal radii. Both circles touch one leg and the hypotenuse, and both 
are tangent to each other at one point. How can I find the radius of 
these circles?  

Thanks,
- Dan

Date: 11/04/2002 at 09:20:32
From: Doctor Floor
Subject: Re: Geometry

Hi, Dan,

Thanks for your question.

Let AB be the hypotenuse of the 3,4,5-triangle ABC, I its incenter, 
and I1 and I2 the two centers of the circles you describe. I1 and I2 
must lie on the angle bisectors of angles A and B. We choose I1 in 
such a way that it lies on the angle bisector of A, and I2 lies on the 
angle bisector of B. Let R be the radius of the two little circles.

We know that the inradius of the triangle is 1, using for instance:

   Radius of a Circle Inscribed in a Triangle
   http://mathforum.com/library/drmath/view/55138.html    

                   C
               _.-' \
           _.-'      \
       _.-'      I    \
   _.-'   I2        I1 \
 A----------------------B

Now we see that triangles ABI and I1I2I must be similar, since AB and 
I1I2 are clearly parallel. The altitude to AB in triangle ABI is 
clearly the inradius of ABC, and we have seen that it is 5. The 
altitude to I1I2 in triangle I1I2I has length 1-R, while I1I2 has 
length 2R. From the similarity we see:

 5 : 1 = 2R : 1-R

and thus

 5-5R = 2R
 5 = 7R
 R = 5/7

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/