KöMaL 2011 A544

A circle $k$ with center $O$ and four distinct fixed points $A$, $B$, $C$, $D$ lying on it are given. The circle $k$' intersects $k$ perpendicularly at $A$ and $B$. Let $X$ be a variable point on the line $OA$. Let $U$, other than $A$, be the second intersection of the circles ACX and $k$'. Let $V$, other than $A$, be the second intersection of the circles ADX and $k$'. Let $W$, other than $B$, be the second intersection of the circle BDU and the line $OB$. Finally, let $E$, other than $B$, be the second intersection of the circles BVW and $k$. Prove that the location of the point $E$ is independent from the choice of the point $X$.