CN-TST 2023 Problem 16

Let $\Gamma, \Gamma_1, \Gamma_2$ be mutually tangent circles. The three circles are also tangent to a line $l$ . Let $\Gamma, \Gamma_1$ be tangent to each other at $B_1$ , $\Gamma, \Gamma_2$ be tangent to each other at $B_2$ , $\Gamma_1, \Gamma_2$ be tangent to each other at $C$ . $\Gamma, \Gamma_1, \Gamma_2$ are tangent to $l$ at $A, A_1, A_2$ respectively, where $A$ is between $A_1,A_2$ . Let $D_1 = A_1C \cap A_2B_2, D_2 = A_2C \cap A_1B_1$ . Prove that $D_1D_2$ is parallel to $l$ .