Question 4
Let $ABC$ be an acute triangle and let $H$ be its orthocenter. A point $G$ lies in the plane of the triangle such that $ABGH$ is a parallelogram. A point $I$ lies on the line $GH$ such that the line $AC$ bisects the segment $HI$. The line $AC$ intersects the circumcircle of triangle $GCI$ for a second time at the point $J$. Prove that $IJ = AH$.