SHARYGIN 2023 Problem 6

Let $A_1, B_1, C_1$ be the feet of altitudes of an acute-angled triangle $ABC$ . The incircle of triangle $A_1B_1C_1$ touches $A_1B_1, A_1C_1, B_1C_1$ at points $C_2, B_2, A_2$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ concur at a point lying on the Euler line of triangle $ABC$ .