SHARYGIN 2011 Problem 8

The incircle of right-angled triangle $ABC$ ($\angle B = 90^\circ$) touches $AB, BC, CA$ at points $C_1, A_1, B_1$ respectively. Points $A_2, C_2$ are the reflections of $B_1$ in lines $BC, AB$ respectively. Prove that lines $A_1A_2$ and $C_1C_2$ meet on the median of triangle $ABC$.