KöMaL 2012 B4448

The escribed circle drawn to side $AC$ of a triangle ABC touches the lines of sides $BC$, $AC$ and $AB$ at the points $A_1$, $B_1$ and $C_1$, respectively. Let $F$ denote the midpoint of the line segment $A_1 B_1$. Prove that $\angle B_1 C_1 C$=$\angle A_1 C_1 F$.