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May 1, 2016
Posted by **elisabeth** on Monday, November 28, 2011 at 11:36am.

Prove that this relationship is true for the inscribed circle in any right triangle.

- geometry -
**Steve**, Monday, November 28, 2011 at 12:33pmIn the triangle ABC call the incenter point P.

The radius r of the incircle is just the altitude from P to each side of the triangle. Let a,b,c be the sides opposite A,B,C, respectively.

Draw a line from P to each vertex, A,B,C. This divides the triangle into three smaller triangles, each with height r.

So, the area of ABC is just r/2 (a+b+c)

Or, where p is the perimeter of ABC:

A = r/2 * p

r = 2A/p

This is true for all triangles, including right triangles.