# Inradius

The **inradius** of a polygon is the radius of its incircle (assuming an incircle exists). It is commonly denoted .

In a triangle, the incenter is where the three angle bisectors meet.

## A Property

- If has inradius and semi-perimeter , then the area of is . This formula holds true for other polygons if the incircle exists.

# Proof

Add in the incircle and drop the altitudes from the incenter to the sides of the triangle. Also draw the lines , and . After this AB, AC, and BC are the bases of , and respectively. But they all have the same height(the inradius), so .

Also the inradius of a incircle inscribed in a right triangle is as by drawing three inradiuses to the three tangent points, then A to that tangent point is equal to A to the other tangent point (explained in circles) and etc for B and C. After doing it for B and C, C (the hypotenuse) should equal the equivalent tangent of A and the equivalent tangent of B added together, thus our equation A = rs

## Problems

- Verify the inequality .
- Verify the identity (see Carnot's Theorem).
- 2007 AIME II Problems/Problem 15