Hello, Ultros88!

If the diagonals of a quadrilateral bisect each other,

then the quadrilateral is a parallelogram. (vector proof) Code:

A B
* - - - - - - - - *
/ * * /
/ * * /
/ * /
/ * E * /
/ * * /
* - - - - - - - - *
D C

We are given: .$\displaystyle \begin{array}{cccc}\overrightarrow{AE} &=& \overrightarrow{EC} & {\color{blue}[1]}\\ \overrightarrow{EB} &=& \overrightarrow{DE} & {\color{blue}[2]}\end{array}$

We see that: .$\displaystyle \begin{array}{cccc}\overrightarrow{AB} &=& \overrightarrow{AE} + \overrightarrow{EB} & {\color{blue}[3]}\\ \overrightarrow{DC} &=& \overrightarrow{DE} + \overrightarrow{EC} & {\color{blue}[4]} \end{array}$

Substitute [1] and [2] into [3]: .$\displaystyle \overrightarrow{AB} \:=\:\overrightarrow{EC} + \overrightarrow{DE}$

. . Hence: .$\displaystyle \overrightarrow{AB} \:=\:\overrightarrow{DC}$

Since $\displaystyle \overrightarrow{AB} \parallel \overrightarrow{DC}\text{ and }|\overrightarrow{AB}| = |\overrightarrow{DC}|$, then $\displaystyle ABCD$ is a parallelogram.

Theorem: If two sides of a quadrilateral are parallel and equal,

. . . . . . . the quadrilateral is a parallelogram.

.