SHARYGIN 2013 Problem 16

The incircle of triangle $ABC$ touches $BC$ , $CA$ , $AB$ at points $A_1$ , $B_1$ , $C_1$ , respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$ . Prove that $CK$ is parallel to $AB$ .