MathLabs

Problem 3

A circle is inscribed in a triangle ABCABC with sides a,b,ca,b,c. Tangents to the circle parallel to the sides of the triangle are constructed. Each tangent cuts off a triangle from △ABC\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,ca,b,c.
Step 2 of 5: Find a corner area ratio
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[BHM][ABC]=(a+c−bp)2\frac{[BHM]}{[ABC]}=\left(\frac{a+c-b}{p}\right)^2
Detailed analysis

For the corner triangle at BB, the parallel tangent makes a triangle similar to ABCABC. Its perimeter is a+c−ba+c-b, so its linear scale is (a+c−b)/p(a+c-b)/p and its area ratio is the displayed square. The other two corner triangles give cyclic analogues.