Let $E$ and $F$ be points on the side $AB$ of a rectangle $ABCD$ such that $AE = EF$. The line through $E$ perpendicular to $AB$ intersects the diagonal $AC$ at $G$, and the segments $FD$ and $BG$ intersect at $H$. Prove that the areas of the triangles $FBH$ and $GHD$ are equal.