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 5 of 5: Simplify the result
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16πΔ2(a2+b2+c2)(a+b+c)4\boxed{\frac{16\pi\Delta^2(a^2+b^2+c^2)}{(a+b+c)^4}}
Detailed analysis

The identity p2+(a+c−b)2+(a+b−c)2+(b+c−a)2=4(a2+b2+c2)p^2+(a+c-b)^2+(a+b-c)^2+(b+c-a)^2=4(a^2+b^2+c^2) gives the required sum 16πΔ2(a2+b2+c2)/(a+b+c)4\boxed{16\pi\Delta^2(a^2+b^2+c^2)/(a+b+c)^4}.