# Equation for Perpendicular Bisector of Two Points

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## Theorem

Let $\tuple {x_1, y_1}$ and $\tuple {y_1, y_2}$ be two points in the cartesian plane.

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:

- $y - \dfrac {y_1 + y_2} 2 = \dfrac {x_1 - x_2} {y_2 - y_1} \paren {x - \dfrac {x_1 + x_2} 2}$

## Proof

Let $M$ be the straight line passing through $z_1$ and $z_2$.

Let $Q$ be the midpoint of $M$.

We have that:

- $Q = \tuple {\dfrac {x_1 + x_2} 2, \dfrac {y_1 + y_2} 2}$

The slope of $M$ is $\dfrac {y_2 - y_1} {x_2 - x_1}$.

As $L$ is perpendicular to the $M$, its slope is $\dfrac {x_1 - x_2} {y_2 - y_1}$.

Thus by Equation of Straight Line in Plane: Point-Slope Form, the equation for $L$ is:

- $y - \dfrac {y_1 + y_2} 2 = \dfrac {x_1 - x_2} {y_2 - y_1} \paren {x - \dfrac {x_1 + x_2} 2}$

$\blacksquare$