Equation for Perpendicular Bisector of Two Points in Complex Plane

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.

Parametric Form $1$

$L$ 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.

Parametric Form $2$

Equation for Perpendicular Bisector of Two Points in Complex Plane/Parametric Form 2

Standard Form

$L$ can be expressed by the equation:

$\map \Re {z_2 - z_1} x + \map \Im {z_2 - z_1} y = \dfrac {\cmod {z_2}^2 - \cmod {z_1}^2} 2$

Examples

Perpendicular Bisector of $2 + i$ and $3 - 2 i$

Let $L$ be $L$ be the perpendicular bisector of the straight line through $2 + i$ and $3 - 2 i$ in the complex plane.

Then $L$ can be expressed as an equation in the following ways:

Parametric Form: $1$

$z - \paren {\dfrac 5 2 - \dfrac i 2} = t \paren {3 + i}$

Parametric Form: $2$

$x = 3 t + \dfrac 5 2, y = t - \dfrac 1 2$

Standard Form

$x - 3 y = 4$