BD-BDMO 2018 Problem 5

Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $E$ . The two circles passing through $B$ meet again at $F$ . The two circles passing through $C$ meet again at $G$ . The two circles passing through $D$ meet again at $H$ . Suppose, $E, F, G,H$ are all distinct. Is the quadrilateral $EFGH$ similar to $ABCD$ ? Show with proof.