Question 1
Let $ABC$ be an acute triangle and $O$ be its circumcircle. Let $O'$ be the circle that is tangent to $O$ at $A$ and tangent to the side $BC$ at $D$. $O'$ intersects the lines $AB$ and $AC$ at $E$ and $F$, respectively. $O'$ intersects the lines $OO'$ and $EO'$ at $A' (\neq A)$ and $G (\neq E)$, respectively. The lines $BO$ and $A'G$ intersect at $H$. Prove that $DF^2 = AF \cdot GH$.