Question 56
Let $A$, $B$, $C$ and $D$ be points on the circle $\omega$ such that $ABCD$ is a convex quadrilateral. Suppose that $AB$ and $CD$ intersect at a point $E$ such that $A$ is between $B$ and $E$ and that $BD$ and $AC$ intersect at a point $F$. Let $X \neq D$ be the point on $\omega$ such that $DX$ and $EF$ are parallel. Let $Y$ be the reflection of $D$ through $EF$ and suppose that $Y$ is inside the circle $\omega$.
Show that $A$, $X$, and $Y$ are collinear.
Show that $A$, $X$, and $Y$ are collinear.