Question 53
Let $ABC$ be a scalene and acute triangle, with circumcentre $O$. Let $\omega$ be the circle with centre $A$, tangent to $BC$ at $D$. Suppose there are two points $F$ and $G$ on $\omega$ such that $FG \perp AO$, $\angle BFD = \angle DGC$ and the couples of points $(B, F)$ and $(C, G)$ are in different halfplanes with respect to the line $AD$. Show that the tangents to $\omega$ at $F$ and $G$ meet on the circumcircle of $ABC$.