Question

Prove that the midpoints of the sides of a convex quadrangle are the vertices of a parallelogram.

Answer 1

The lengths of sides of the quadrilateral formed by mid points is $\frac{1}{2}(\overrightarrow{A}+\overrightarrow{B}),\frac{1}{2}(\overrightarrow{B}+\overrightarrow{C}),\frac{1}{2}(\overrightarrow{C}+\overrightarrow{D}),\frac{1}{2}(\overrightarrow{D}+\overrightarrow{A})$ in that order

Since the quadrilateral is closed, we have the following identities.

$\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{C}+\overrightarrow{D}$

and

$\overrightarrow{A}+\overrightarrow{C}=\overrightarrow{B}+\overrightarrow{D}$

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

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

This shows that the opposites sides of the quadrilateral formed by joining the midpoints have same lengths. Thus it must be a parallelogram.