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## Q. 3.1

Show that the midpoints of the sides of a quadrilateral are the vertices of a parallelogram.

## Verified Solution

The situation is sketched in Figure 1 . Using the fact that $B, D, F$, and $H$ are midpoints, we have

$\overrightarrow{B D}=\overrightarrow{B C}+\overrightarrow{C D}=\frac{1}{2} \overrightarrow{A C}+\frac{1}{2} \overrightarrow{C E}=\frac{1}{2}(\overrightarrow{A C}+\overrightarrow{C E})=\frac{1}{2} \overrightarrow{A E}$

and

$\overrightarrow{H F}=\overrightarrow{H G}+\overrightarrow{G F}=\frac{1}{2} \overrightarrow{A G}+\frac{1}{2} \overrightarrow{G E}=\frac{1}{2}(\overrightarrow{A G}+\overrightarrow{G E})=\frac{1}{2} \overrightarrow{A E}$

Thus $\overrightarrow{B D}=\overrightarrow{H F}$. Similarly, $\overrightarrow{H B}=\overrightarrow{F D}$, so $B D F H$ is a parallelogram.