Quadrilateral is Parallelogram iff Diagonals Bisect each other

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Theorem

Let $ABCD$ be a quadrilateral.

Then:

$ABCD$ is a parallelogram

if and only if:

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


Proof

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 $AD$.

$\Box$


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.

$\blacksquare$


Sources