CH-SMO 2010 Problem 9

Let $k$ and $k'$ two concentric circles centered at $O$ , with $k'$ being larger than $k$ . A line through $O$ intersects $k$ at $A$ and $k'$ at $B$ such that $O$ seperates $A$ and $B$ . Another line through $O$ intersects $k$ at $E$ and $k'$ at $F$ such that $E$ separates $O$ and $F$ . Show that the circumcircle of $\triangle{OAE}$ and the circles with diametres $AB$ and $EF$ have a common point.