# 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 $AD$.

## Proof

### Sufficient Condition

Let $ABCD$ be a parallelogram.

$AD$ is a bisector of $BC$

and

$BC$ is a bisector of $AD$.

$\Box$

### 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.

$\blacksquare$