KZ-MO 1999 Problem 3

The circle inscribed in the triangle $ABC$ , with center $O$ , touches the sides $AB$ and $BC$ at the points $C_1$ and $A_1$ , respectively. The lines $CO$ and $AO$ intersect the line $C_1A_1$ at the points $K$ and $L$ . $M$ is the midpoint of $AC$ and $\angle ABC = 60^\circ$ . Prove that $KLM$ is a regular triangle.