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.

Midpoints of Quadrilaterals

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

Theorem

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

Given:

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

What To Show

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

Proof

Draw $latex BD$.

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

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

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

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

Draw $latex AC$.

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

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

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

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

 

 

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