Quadrilateral is Parallelogram iff Diagonals Bisect each other

Theorem
Let $ABCD$ be a quadrilateral.

Then:
 * $ABCD$ is a parallelogram


 * both:
 * $AD$ is a bisector of $BC$
 * and:
 * $BC$ is a bisector of $AD$.
 * $BC$ is a bisector of $AD$.

Sufficient Condition
Let $ABCD$ be a parallelogram.

Then by Diameters of Parallelogram Bisect each other:
 * $AD$ is a bisector of $BC$

and
 * $BC$ is a bisector of $AD$.

Necessary Condition
Let $ABCD$ be such that:


 * $AD$ is a bisector of $BC$

and
 * $BC$ is a bisector of $AD$.

Then from Quadrilateral with Bisecting Diagonals is Parallelogram, $ABCD$ is a parallelogram.