SHARYGIN 2013 Problem 20

Let $C_1$ be an arbitrary point on the side $AB$ of triangle $ABC$ . Points $A_1$ and $B_1$ on the rays $BC$ and $AC$ are such that $\angle AC_1B_1 = \angle BC_1A_1 = \angle ACB$ . The lines $AA_1$ and $BB_1$ meet in point $C_2$ . Prove that all the lines $C_1C_2$ have a common point.