I chanced upon this interesting theorem which states the following:
On the sides of parallelogram $ABCD$, squares are constructed exterior to it. Then, their centers, $M_1, M_2, M_3 \text{ and } M_4 $ are themselves vertices of a square. 
Also, does anyone knows of similar interesting theorems which I may prove using complex numbers? Thank you!
In a more general context, that's a consequence of the Napoleon-Barlotti theorem