OLYMPIC-REVENGE 2020 Problem 3

Let $ABC$ be a triangle and $\omega$ its circumcircle. Let $D$ and $E$ be the feet of the angle bisectors relative to $B$ and $C$ , respectively. The line $DE$ meets $\omega$ at $F$ and $G$ . Prove that the tangents to $\omega$ through $F$ and $G$ are tangents to the excircle of $\triangle ABC$ opposite to $A$ .