Perpendicular bisector ⇔ equidistant from endpoints

Dependencies: Linear pair sums to 180°, SAS congruence, SSS congruence.

Statement

Let $C$ and $D$ be two distinct points in the plane. The perpendicular bisector $\ell$ of segment $CD$ is defined as the line passing through the midpoint $M$ of $CD$ and perpendicular to the line $CD$.

The theorem gives the point-set characterization of $\ell$: for any point $P$ in the plane,

In other words, $\ell$ is exactly the set of points "equidistant from $C$ and $D$" — this is the first locus theorem in middle-school geometry that recharacterizes "a line" via a distance condition.