SHARYGIN 2009 Problem 16

Three lines passing through point $O$ form equal angles by pairs. Points $A_1$ , $A_2$ on the first line and $B_1$ , $B_2$ on the second line are such that the common point $C_1$ of $A_1B_1$ and $A_2B_2$ lies on the third line. Let $C_2$ be the common point of $A_1B_2$ and $A_2B_1$ . Prove that angle $C_1OC_2$ is right.