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 $latex QM \parallel NP$. Continue reading

The Proof of the Midsegment Theorem

The segment connecting the two midpoints of the sides of a triangle is called its midsegment or midline. The midsegment of a triangle has interesting characteristics:

(1) the midsegment connecting the midpoints of the two sides of a triangle is parallel to the third side and (2) its length is also half of the third side. In this post, we prove these two theorems.

midsegment

In the given above, $latex PQR$ is a triangle and $latex MN$ is a midsegment. Continue reading