OFM 2022 S3

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Denote $\Delta$ the tangent at $A$ to the circle $\Gamma$ . $\Gamma_1$ is a circle tangent to the lines $\Delta$ , $(AB)$ and $(BC)$ , and $E$ its touchpoint with the line $(AB)$ . Let $\Gamma_2$ be a circle tangent to the lines $\Delta$ , $(AC)$ and $(BC)$ , and $F$ its touchpoint with the line $(AC)$ . We suppose that $E$ and $F$ belong respectively to the segments $[AB]$ and $[AC]$ , and that the two circles $\Gamma_1$ and $\Gamma_2$ lie outside triangle $ABC$ . Show that the lines $(BC)$ and $(EF)$ are parallel.