IMO 2010 Problem 2

Given a triangle $ABC$ , with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$ . Let $E$ be a point on the arc $BDC$ , and $F$ a point on the segment $BC$ , such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$ . If $G$ is the midpoint of $IF$ , prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$ .

Proposed by Tai Wai Ming and Wang Chongli, Hong Kong