DE-BUNDESWETTBEWERB 2012 Problem 7

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ in points $A_1, B_1$ and $C_1$ respectively. $C_1D$ is a diameter of the incircle. Finally, let $E$ be the intersection of the lines $B_1C_1$ and $A_1D$ . Prove that the segments $CE$ and $CB_1$ have equal length.