1964 IMO 第 3 题

The incircle is inscribed in a triangle $A B C$ with sides $a, b, c$. Three tangents to the incircle are drawn, each of which is parallel to one side of the triangle $A B C$. These tangents form three smaller triangles (internal to $\triangle A B C$ ) with the sides of $\triangle A B C$. In each of these triangles an incircle is inscribed. Determine the sum of areas of all four incircles.

内切圆内切于边为 $a、b、c$ 的三角形 $A B C$。绘制内切圆的三条切线，每条切线均平行于三角形 $A B C$ 的一条边。这些切线与 $\triangle A B C$ 的边形成三个较小的三角形($\triangle A B C$ 的内部)。在每个三角形中都有一个内切圆。确定所有四个内切圆的面积之和。