SHARYGIN 2025 Problem 108

Let $ABC$ be an acute-angled triangle. A line $l$ meets the sides $AB, AC$ and the sideline $BC$ at points $C_1,B_1,A_1$ respectively. A circle $\omega_a$ touches $BC$ at $A_1$ , and touches the minor arc $BC$ of the circumcircle of $ABC$ . Circles $\omega_b$ , $\omega_c$ are defined similarly. Prove that these three circles have a common tangent.

Proposed by K. Belsky