KöMaL 2017 A690

In a convex quadrilateral $ABCD$, the perpendicular drawn from $A$ to line $BC$ meets the lines $BC$ and $BD$ at $P$ and $U$, respectively. The perpendicular drawn from $A$ to line $CD$ meets the lines $CD$ and $BD$ at $Q$ and $V$, respectively. The midpoints of the segments $BU$ and $DV$ are $S$ and $R$, respectively. The lines $PS$ and $QR$ meet at $E$. The second intersection point of the circles $PQE$ and $RSE$, other than $E$, is $M$. The points $A$, $B$, $C$, $D$, $E$, $M$, $P$, $Q$, $R$, $S$, $U$, $V$ are distinct. Show that the center of the circle $BCD$, the center of the circle $AUV$ and the point $M$ are collinear.