Quadrilateral with Bisecting Diagonals is Parallelogram

Theorem
Let $ABCD$ be a quadrilateral.

Let the diagonals of $ABCD$ bisect each other.

Then $ABCD$ is a parallelogram.

Proof
The diagonals of $ABCD$ bisect each other if the position vectors of the midpoints of the diagonals are the same point.

Let $z_1, z_2, z_3, z_4$ be the position vectors of the vertices of $ABCD$.

Thus:

The result follows from Condition for Points in Complex Plane to form Parallelogram.

Also see

 * Diameters of Parallelogram Bisect each other