JBMO-SL 2008 G10

Let $\Gamma$ be a circle of center $O$, and $\delta$ be a line in the plane of $\Gamma$, not intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto $\delta$, and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$ the circle of diameter $A M$, by $X$ the (other than $M$ ) intersection point of $\gamma$ and $\Gamma$, and by $Y$ the (other than $A$ ) intersection point of $\gamma$ and $\delta$. Prove that the line $X Y$ passes through a fixed point.