Question 186
Let $A_{1}, B_{1}, C_{1}$ be points in the interior of sides $BC, CA, AB$, respectively, of equilateral triangle $ABC$. Prove that if the radii of the inscribed circles of $\triangle C_{1}AB_{1}$, $\triangle B_{1}CA_{1}$, $\triangle A_{1}BC_{1}$, $\triangle A_{1}B_{1}C_{1}$ are equal, then $A_{1}, B_{1}, C_{1}$ are the midpoints of the sides of $\triangle ABC$ on which they lie.