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 $latex ABCD$ 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 $latex QM \parallel NP$. Continue reading →