Diameters of Parallelogram Bisect each other/Proof 2

Proof

 * DiametersOfParallelogramBisect-Complex.png

Let $\Box ABCD$ be embedded in the complex plane such that $B$ is identified with the origin $0 + 0 i$.

Let $A$ be identified with the complex number $z_1$.

Let $C$ be identified with the complex number $z_2$.

By Geometrical Interpretation of Complex Subtraction:
 * $z_1 - z_2 = AC$

Then:
 * $AE = m \paren {z_1 - z_2}$

for some $m$ where $0 \le m \le 1$.

Similarly, by Geometrical Interpretation of Complex Addition:
 * $z_1 + z_2 = BD$

Then:
 * $BE = n \paren {z_1 + z_2}$

for some $n$ where $0 \le n \le 1$.

Then:

and so $E$ is at the midpoint of the diameters of $\Box ABCD$.