# The Midpoints of the Sides of a Quadrilateral Form a Parallelogram

In the previous post, we have learned that the segment connecting the two sides of the triangle is parallel to the third side and half its length. In this post, we are going to use the said theorem to prove that the midpoints of any quadrilateral determine a parallelogram.

Given quadrilateral $ABCD$ and midpoints $M$, $N$, $P$, $Q$ as shown as above, $MNPQ$ is a parallelogram. To show that $MNPQ$ is a parallelogram, we have to show that their opposite sides are parallel (definition).  That is, we have to show that $MN \parallel PQ$ and $QM \parallel NP$.

Theorem

The quadrilateral formed by joining the consecutive midpoints of another quadrilateral is a parallelogram.

Given:

Quadrilateral $ABCD$. $M,$latex N\$, $N$, $Q$ are midpoints of $AD$, $AB$, $BC$ and $CD$ respectively.

What To Show

$MN \parallel PQ$ and $QM \parallel NP$

Proof

Draw $BD$.

$MN$ is the midsegment of triangle $ABD$, so $MN$ is parallel to $BD$.

$QP$ is also a midsegment of triangle $BCD$, so $QP$ is parallel to $BD$.

$MN$ is parallel to $BD$ and $BD$ is parallel to $QP$, by transitivity, $MN$ is parallel to $QP$.

It remains to show that $MQ$ is parallel to $NP$.

Draw $AC$.

$MQ$ is a midsegment of triangle $ACD$, so $MQ$ is parallel to $AC$.

$NP$ is a midsegment of triangle $ABC$, so $NP$ is parallel to $AC$.

Since $MQ$ is parallel to $AC$ and $AC$ is parallel to $NP$, by transitivity, $MQ$ is parallel $NP$.

Since $MN$ is parallel to $QP$, and $MQ$ is parallel to $NP$, $MNPQ$ is a parallelogram (by definition).