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 3 of 5: Compute its inradius

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r_B=\frac{2\Delta(a+c-b)}{p^2}

Detailed analysis

The corner triangle has area $\Delta(a+c-b)^2/p^2$ and perimeter $a+c-b$ . Since inradius equals twice area divided by perimeter, its radius is $r_B$ as displayed.