Question 91
For the quadrilateral $ABCD$, let $AC$ and $BD$ intersect at $E$, $AB$ and $CD$ intersect at $F$, and $AD$ and $BC$ intersect at $G$. Additionally, let $W, X, Y$, and $Z$ be the points of symmetry to $E$ with respect to $AB, BC, CD$, and $DA$ respectively. Prove that one of the intersection points of $\odot(FWY)$ and $\odot(GXZ)$ lies on the line $FG$.