IMO 1988 LL MON63

Origin: MON

Problem

Let ABCD be a quadrilateral. Let A′BCD′ be the reflection of ABCD in BC, while A′′B′CD′ is the reflection of A′BCD′ in CD′ and A′′B′′C′D′ is the reflection of A′′B′CD′ in D′A′′. Show that if the lines AA′′ and BB′′ are parallel, then ABCD is a cyclic quadrilateral.