SHARYGIN 2022 Problem 4

Let $AA_1$ , $BB_1$ , $CC_1$ be the altitudes of acute angled triangle $ABC$ . $A_2$ be the touching point of the incircle of triangle $AB_1C_1$ with $B_1C_1$ , points $B_2$ , $C_2$ be defined similarly. Prove that the lines $A_1A_2$ , $B_1B_2$ , $C_1C_2$ concur.