Radius of a Circle Inscribed in a Triangle

Date: 06/02/99 at 19:51:38
From: Michael
Subject: Radius of an inscribed circle

There is a drawing of a right triangle with sides of lengths 3, 4, and 
5. A circle is inscribed in the triangle so that each side is tangent 
to the circle. The question is: What is the radius of an inscribed 
circle of a triangle with sides 3, 4, and 5?

I have tried a few different methods, none of which seems to work. 
What should I do?

Date: 06/03/99 at 08:03:23
From: Doctor Floor
Subject: Re: Radius of an inscribed circle

Hi, Michael,

Thanks for your question!

Let's consider the general situation of a right triangle:

     B
     |  \
     |     Z            I is the center of the inscribed circle.
     X   I    \         X,Y and Z are the points where the inscribed 
     |           \      circle meets the sides, so XI = YI = ZI = r.
     C---Y----------A

I write a = BC (length), b = AC and c = AB.

We know that area(ABC) = 0.5*a*b   .........................[1].

Also area(ABC) = area(ABI) + area(CIA) + area(BCI)
              = 0.5*r*a + 0.5*r*b + 0.5*r*c
              = 0.5*r*(a+b+c)      .........................[2].

From [1] = [2] we find the formula:

  r = a*b/(a+b+c).

I suppose you can do the rest yourself now. I hope this helps you out.

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