KöMaL 2011 B4348

The diagonals of a cyclic quadrilateral ABCD are not perpendicular. The feet of the perpendiculars dropped from the vertices $A$, $B$, $C$, $D$ onto the diagonals not passing through them are $A$', $B$', $C$', $D$', respectively. The intersections of lines $AA$' and $DD$', $DD$' and $CC$', $CC$' and $BB$', and finally, $BB$' and $AA$' are $E$, $F$, $G$, $H$, respectively. Prove that $A$'$B$'$C$'$D$' is a cyclic quadrilateral with the centre of its circumscribed circle lying at the intersection of the line segments $EG$ and $FH$. (Suggested by B. Bíró, (Eger)