MEMO 2014 I3

Let $ABC$ be a triangle with $AB < AC$ and incentre $I$ . Let $E$ be the point on the side $AC$ such that $AE = AB$ . Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$ .

Prove that the line $AI$ , the line perpendicular to $AE$ at $E$ , and the bisector of the angle $\angle BGI$ are concurrent.