Problem 3

A circle is inscribed in a triangle

ABC

with sides

a,b,c

. Tangents to the circle parallel to the sides of the triangle are constructed. Each tangent cuts off a triangle from

\triangle ABC

, and a circle is inscribed in each of these three triangles. Find the sum of the areas of all four inscribed circles in terms of

a,b,c

Step 1 of 5: Set the perimeter notation

p=a+b+c

Detailed analysis

Let $\Delta$ be the area of $ABC$ and put $p=a+b+c$ . The original incircle has radius $2\Delta/p$ and area $4\pi\Delta^2/p^2$ .