OMM 1990 Problem 6

$ABC$ is a triangle with $\angle C = 90^o$ . $E$ is a point on $AC$ , and $F$ is the midpoint of $EC$ . $CH$ is an altitude. $I$ is the circumcenter of $AHE$ , and $G$ is the midpoint of $BC$ . Show that $ABC$ and $IGF$ are similar.