Condition for Points in Complex Plane to form Parallelogram

Theorem
Let $A = z_1$, $B = z_2, C = z_3$ and $D = z_4$ represent on the complex plane the vertices of a quadrilateral.

Then $ABCD$ is a parallelogram iff:
 * $z_1 - z_2 + z_3 - z_4 = 0$

Proof

 * ParallelogramInComplexPlane.png

$ABCD$ is a parallelogram iff:
 * $\vec {AB} = \vec {DC}$

That is iff:
 * $z_2 - z_1 = z_3 - z_4$

from which:
 * $z_1 - z_2 + z_3 - z_4 = 0$