Question 100
Let $A$, $B$, $C$ and $D$ be points on a circle $\Gamma$ such that the lines $AB$ and $CD$ intersect at the point $T$, where $A$ lies between $B$ and $T$ and $D$ lies between $C$ and $T$. Let the line through $D$ that is parallel to the line $AB$ intersect the circle $\Gamma$ again at the point $E$, and let the line $ET$ intersect $\Gamma$ again at the point $F$. Let $G$ denote the point of intersection of the lines $CF$ and $AB$. Let $X$ be the midpoint of the line segment $AB$ and let $Y$ be the point that we obtain by reflecting the point $T$ through the point $G$. Prove that the points $X$, $Y$, $C$ and $D$ are concyclic.
