ABCDE is a pentagon. Squares have been constructed on each of the 4 sides of this pentagon acting as a base. The centroid of each square has been determined and a pentagon constructed with these centre points as vertices as shown in the diagrom that is attached.

I would like to prove that given that the smaller pentagon ABCDE is REGULAR this implies the larger penatgon PQRST is also a regular pentagon. Can anyone help?