SHARYGIN 2023 Problem 93

Points $A_1$ , $A_2$ , $B_1$ , $B_2$ lie on the circumcircle of a triangle $ABC$ in such a way that $A_1B_1 \parallel AB$ , $A_1A_2 \parallel BC$ , $B_1B_2 \parallel AC$ . The line $AA_2$ and $CA_1$ meet at point $A'$ , and the lines $BB_2$ and $CB_1$ meet at point $B'$ . Prove that all lines $A'B'$ concur.