Question 2
Let $I$ be the incenter of a triangle $ABC$ and let $\Gamma$ be its circumcircle. Let line $AI$ intersect $\Gamma$ again at $D$. Let $E$ be a point on arc $\widehat{BDC}$ and $F$ a point on side $BC$ such that $$
\angle BAF = \angle CAE \lt \tfrac12 \angle BAC.
$$ Finally, let $G$ be the midpoint of $\overline{IF}$. Prove that $\overline{DG}$ and $\overline{EI}$ intersect on $\Gamma$.
\angle BAF = \angle CAE \lt \tfrac12 \angle BAC.
$$ Finally, let $G$ be the midpoint of $\overline{IF}$. Prove that $\overline{DG}$ and $\overline{EI}$ intersect on $\Gamma$.