HMMT 2021 T2

Let $A B C$ be a right triangle with $\angle A=90^{\circ}$. A circle $\omega$ centered on $B C$ is tangent to $A B$ at $D$ and $A C$ at $E$. Let $F$ and $G$ be the intersections of $\omega$ and $B C$ so that $F$ lies between $B$ and $G$. If lines $D G$ and $E F$ intersect at $X$, show that $A X=A D$.