SHARYGIN 2023 Problem 17

A common external tangent to circles $\omega_1$ and $\omega_2$ touches them at points $T_1, T_2$ respectively. Let $A$ be an arbitrary point on the extension of $T_1T_2$ beyond $T_1$ , and $B$ be a point on the extension of $T_1T_2$ beyond $T_2$ such that $AT_1 = BT_2$ . The tangents from $A$ to $\omega_1$ and from $B$ to $\omega_2$ distinct from $T_1T_2$ meet at point $C$ . Prove that all nagelians of triangles $ABC$ from $C$ have a common point.