Equation for Perpendicular Bisector of Two Points in Complex Plane/Parametric Form 1

Theorem
Let $z_1, z_2 \in \C$ be complex numbers.

Let $L$ be the perpendicular bisector of the straight line through $z_1$ and $z_2$ in the complex plane.

$L$ can be expressed by the equation:
 * $z = ...$

or:
 * $z = ...$

This form of $L$ is known as the parametric form, where $t$ is the parameter.

Proof
Let $z_1$ and $z_2$ be represented by the points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the complex plane.

Let $z$ be an arbitrary point on $L$ represented by the point $P$.


 * Perpendicular Bisector of Two Points in Complex Plane.png

We have that $L$ passes through the point:
 * $\dfrac {z_1 + z_2} 2$

and is perpendicular to the straight line:
 * $z = z_1 + t \paren {z_2 - z_1}$