SHARYGIN 2012 Problem 8

Let $BM$ be the median of right-angled triangle $ABC$ ($\angle B = 90^\circ$). The incircle of triangle $ABM$ touches sides $AB, AM$ in points $A_1, A_2$; points $C_1, C_2$ are defined similarly. Prove that lines $A_1A_2$ and $C_1C_2$ meet on the bisector of angle $ABC$.