Next construct a smaller square (A'B'C'D') inside ABCD, with any orientation and centre and join the corresponding corners.

This divides the region between the squares into four.

We will name these divisions (and their areas):

(N)orth = ABB'A'
(E)ast = BCC'B'
(S)outh = CDD'C'
(W)est = DAA'D'.

Show that the areas of these regions satisfy the equality
N+S = E+W.

In your example (with both ABCD and A'B'C'D' labeled CCW,
AB parallel to A'B', and both AB and A'B' at the top):

Using the complex cross product, the area of the North
region is negative and the area of the East, West, and
South regions are positive. So that N+S = E+W.

Note: In my Solution post change every place you see "2i*"
      to "4i*".