Midpoints of Sides of Quadrilateral form Parallelogram

Theorem

Let $\Box ABCD$ be a quadrilateral.

Let $E, F, G, H$ be the midpoints of $AB, BC, CD, DA$ respectively.

Then $\Box EFGH$ is a parallelogram.

Let $z_1, z_2, z_3, z_4$ be the position vectors of the vertices of $\Box ABCD$.

The midpoints of the sides of $\Box ABCD$ are, then:

$\ds OE$

$=$

$\ds \frac {z_1 + z_2} 2$

$\ds OF$

$=$

$\ds \frac {z_2 + z_3} 2$

$\ds OG$

$=$

$\ds \frac {z_3 + z_4} 2$

$\ds OH$

$=$

$\ds \frac {z_4 + z_1} 2$

Take two opposite sides $EF$ and $HG$:

$\ds EF$

$=$

$\ds \frac {z_2 + z_3} 2 - \frac {z_1 + z_2} 2$

$\ds $

$=$

$\ds \frac {z_3} 2 - \frac {z_1} 2$

$\ds HG$

$=$

$\ds \frac {z_3 + z_4} 2 - \frac {z_4 + z_1} 2$

$\ds $

$=$

$\ds \frac {z_3} 2 - \frac {z_1} 2$

$\ds $

$=$

$\ds EF$

The vectors defining the opposite sides of $\Box EFGH$ are equal.

The result follows by definition of parallelogram.

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $68$