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 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).

 

 

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