The incenter of a triangle is defined by the intersection of the angle bisectors of its three angles. A circle can be inscribed within , as shown in the picture.
This incircle provides a different way of finding the area of commonly needed for high school math contests like the AMC 10/12. Suppose that the sides , , and are known and the inradius is also known. Then is a right triangle with base and height . So
Since the area of is the sum of the areas of these three smaller triangles, we conclude that
where is the semiperimeter of .
This also permits the computation of itself. By Heron’s formula, we know that
Equating these two expressions for the area of , we can solve for the inradius :
For much more about the inradius and incircle, I’ll refer to the MathWorld website.