Take a right angled triangle with sides a, b, c, where c is the hypotenuse.

The incircle is the largest circle that can be drawn inside the triangle, and is tangent to all three sides of the triangle.

Let the radius of the incircle be r.

(a) By writing two different expressions for the perimeter (one should include a, b, c; the other should include a, b, r) find an expression for r.

(b) By writing two different expressions for the area (one should include a, b; the other should include a, b, c, r) find another expression for r.

(c) If you've found the same two expressions that I did, you should be able to equate them and derive a neat proof of a familiar result!