FKMO 1997 Problem 2

The incircle of a triangle $A_1A_2A_3$ is centered at $O$ and meets the segment $OA_j$ at $B_j$ , $j = 1, 2, 3$ . A circle with center $B_j$ is tangent to the two sides of the triangle having $A_j$ as an endpoint and intersects the segment $OB_j$ at $C_j$ . Prove that $$\frac{OC_1+OC_2+OC_3}{A_1A_2+A_2A_3+A_3A_1} \leq \frac{1}{4\sqrt{3}}$$ and find the conditions for equality.