BG-MO 1995 Problem 4

Points $A_1,B_1,C_1$ are selected on the sides $BC$ , $CA$ , $AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$ , $A_1BC_1$ , $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.