We are asked to show the following statement.

If a quadrilateral is a parallelogram, then its diagonals bisect each other .

We are asked to write a coordinate proof, so place the parallelogram in the coordinate plane. It is helpful if you choose a position with one vertex at the origin and one side on the positive

x -

axis. Let's label the coordinates of three of the vertices of the parallelogram.

We can use the properties of a parallelogram to find the coordinates of vertex $C .$

Coordinate	Expression	Justification
$y -$ coordinate	$c$	Opposite sides of the parallelogram are parallel, so points $D$ and $C$ have the same $y -$ coordinates.
$x -$ coordinate	$b + a$	Opposite sides of the parallelogram are congruent, so the distance of points $D$ and $C$ is the same as $A B = a .$

Let's extend the diagram with these coordinates.

We can show that the diagonals bisect each other by finding the midpoints of the two diagonals and show that the two midpoints are the same. Let's use the Midpoint Formula.

The midpoint between points (x_{1}, y_{1}) and (x_{2}, y_{2}) is (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}) .

Let's apply this formula for the endpoints of the two diagonals.

Points	Midpoint
Points	Substitution	Simplification
$A (0, 0)$ and $C (b + a, c)$	$(\frac{0 + ( b + a )}{2}, \frac{0 + c}{2})$	$(\frac{a + b}{2}, \frac{c}{2})$
$B (a, 0)$ and $D (b, c)$	$(\frac{a + b}{2}, \frac{0 + c}{2})$	$(\frac{a + b}{2}, \frac{c}{2})$

We can see that the midpoint of $\overline{A C}$ is the same as the midpoint of $\overline{B D},$ so the diagonals of a parallelogram indeed bisect each other.

Exercise