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:

$z - \dfrac {z_1 + z_2} 2 = i t\paren {z_2 - z_1}$

or:

$z = \dfrac {z_1 + z_2} 2 + i t\paren {z_2 - z_1}$

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 $P$.

We have that $L$ passes through the point:

$\dfrac {z_1 + z_2} 2$

and is perpendicular to the straight line:

$z = z_1 + t \paren {z_2 - z_1}$

By Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle:

the vector $i \paren {z_2 - z_1}$ is perpendicular to the vector $\paren {z_2 - z_1}$

Hence the perpendicular bisector can be written as:

$z = \dfrac {z_1 + z_2} 2 + t i \paren {z_2 - z_1}$

$\blacksquare$