Question 99
Triangle $A B C$ is given. Let $D$ and $E$ be points lying on the rays $C A$ and $C B$, respectively, but not on the sides of triangle $A B C$, such that $|A D|=|B E|=|A B|$. Denote by $G$ the intersection of the lines $A E$ and $B D$, and by $I$ the center of the circle inscribed in triangle $A B C$. Prove that the line $GI$ passes through the midpoint of the side $A B$.