Show that the midpoints of the sides of a quadrilateral are the vertices of a parallelogram.
Question 3.1: Show that the midpoints of the sides of a quadrilateral are ...
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.
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