IrMO 2012 Problem 2

$A,B,C$ and $D$ are four points in that order on the circumference of a circle $K$ . $AB$ is perpendicular to $BC$ and $BC$ is perpendicular to $CD$ . $X$ is a point on the circumference of the circle between $A$ and $D$ . $AX$ extended meets $CD$ extended at $E$ and $DX$ extended meets $BA$ extended at $F$ . Prove that the circumcircle of triangle $AXF$ is tangent to the circumcircle of triangle $DXE$ and that the common tangent line passes through the center of the circle $K$ .