ISL 1988 Problem 52

$ABCD$ is a quadrilateral. $A'BCD'$ is the reflection of $ABCD$ in $BC,$ $A''B'CD'$ 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.