CAPS 2021 Problem 6

Let $ABC$ be an acute triangle and suppose points $A, A_b, B_a, B, B_c, C_b, C, C_a,$ and $A_c$ lie on its perimeter in this order. Let $A_1 \neq A$ be the second intersection point of the circumcircles of triangles $AA_bC_a$ and $AA_cB_a$ . Analogously, $B_1 \neq B$ is the second intersection point of the circumcircles of triangles $BB_cA_b$ and $BB_aC_b$ , and $C_1 \neq C$ is the second intersection point of the circumcircles of triangles $CC_aB_c$ and $CC_bA_c$ . Suppose that the points $A_1, B_1,$ and $C_1$ are all distinct, lie inside the triangle $ABC$ , and do not lie on a single line. Prove that lines $AA_1, BB_1, CC_1,$ and the circumcircle of triangle $A_1B_1C_1$ all pass through a common point.

Josef Tkadlec (Czech Republic), Patrik Bak (Slovakia)