Question 105
Quadrilateral $ABCD$ is both inscribed and circumscribed. Let $E$ be the intersection point of $AD$ and $BC$, $F$ the intersection point of $AB$ and $CD$, $S$ the intersection point of $AC$ and $BD$ and $O$ the circumcenter of quadrilateral $ABCD$. $E'$ and $F'$ are selected on $AB$ and $AD$ such that $\angle BEE' = \angle AEE'$ and $\angle AFF' = \angle DFF'$. Let $M$ be the midpoint of arc $BAD$ of the circumcircle of the quadrilateral and $X$ a point collinear with $O$ and $E'$ such that $\frac{XA}{XB} = \frac{EA}{EB}$. Also let $Y$ be a point collinear with $O$ and $F'$ such that $\frac{YA}{YD} = \frac{FA}{FD}$. Prove that the circle with diameter $OS$, the circumcircle of triangle $OAM$ and the circumcircle of triangle $OXY$ are coaxial.