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.

Outside??  That can't be.  Imagine A'B'C'D' is 90% the size of ABCD and positioned just above ABCD, with the same orientation but completely outside.  ABB'A' (N) and CDD'C' (S) are both large trapezoids.  But BCC'B' (E) and DAA'D' (W) are both narrow slivers.  N+S is clearly much larger than E+W.