Diagonals of a pentagon
 

Let $ABCDE$ be a convex pentagon such that
$$\angle{BAC}=\angle{DBE}, \angle{CBD}=\angle{ECA}, \angle{DCE}=\angle{ADB}, \angle{EDA}=\angle{BEC}, \angle{AEB}=\angle{CAD}$$
and that
$$AB=CD.$$
Show that at least four of its diagonals are of the same length. :great: