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 4 of 5: Sum the four circle areas
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SB=4πΔ2(a+c−b)2p4S_B=4\pi\Delta^2\frac{(a+c-b)^2}{p^4}
Detailed analysis

Squaring rBr_B gives the corner circle area. Add the two cyclic corner terms and the original incircle area: 4πΔ2[p2+(a+c−b)2+(a+b−c)2+(b+c−a)2]/p44\pi\Delta^2\bigl[p^2+(a+c-b)^2+(a+b-c)^2+(b+c-a)^2\bigr]/p^4.