I came up with this but I have not been able to solve it. I would really appreciate any help.
Let $ABC$ be a triangle and let $\omega$ be its circumcircle. Produce the internal angle bisector of $\angle BAC$ to meet $BC$ and $\omega$ in $D$ and $E$, respectively. Let the circle with diameter $DE$ intersect $\omega$ at a second point $F$. Drop a perpendicular from $F$ to "$AC$ produced" and let it meet "$AC$ produced" at $G$. Is it true that line $GF$ is tangent to $\omega$?
(The reason I ask "is it true" is because I don't know if it is.)