Question 125
Let $I$ be the incentre of triangle $ABC$ and let $\Gamma$ be its circumcircle. The line $AI$ meets $\Gamma$ at $A$ and $D$. Choose a point $E$ on the arc $BDC$ and a point $F$ on the side $BC$ such that
$$
\angle BAF = \angle CAE \lt \frac{1}{2} \angle BAC.
$$
Let $G$ be the midpoint of the segment $IF$. Prove that the lines $DG$ and $EI$ intersect at a point on the circle $\Gamma$.
$$
\angle BAF = \angle CAE \lt \frac{1}{2} \angle BAC.
$$
Let $G$ be the midpoint of the segment $IF$. Prove that the lines $DG$ and $EI$ intersect at a point on the circle $\Gamma$.