perpendicular bisector
Let $\overline{AB}$ be a line segment^{} in a plane (we are assuming the Euclidean plane^{}). A bisector^{} of $\overline{AB}$ is any line that passes through the midpoint^{} of $\overline{AB}$. A perpendicular bisector of $\overline{AB}$ is a bisector that is perpendicular^{} to $\overline{AB}$.
It is an easy exercise to show that a line $\mathrm{\ell}$ is a perpendicular bisector of $\overline{AB}$ iff every point lying on $\mathrm{\ell}$ is equidistant from $A$ and $B$. From this, one concludes that the perpendicular bisector of a line segment is always unique.
A basic way to construct the perpendicular bisector $\mathrm{\ell}$ given a line segment $\overline{AB}$ using the standard ruler and compass construction is as follows:

1.
use a compass to draw the circle ${C}_{1}$ centered at point $A$ with radius the length of $\overline{AB}$, by fixing one end of the compass at $A$ and the movable end at $B$,

2.
similarly, draw the circle ${C}_{2}$ centered at $B$ with the same radius as above, with the compass fixed at $B$ and movable at $A$,

3.
${C}_{1}$ and ${C}_{2}$ intersect at two points, say $P,Q$ (why?)

4.
with a ruler, draw the line $\overleftrightarrow{PQ}=\mathrm{\ell}$,

5.
then $\mathrm{\ell}$ is the perpendicular bisector of $\overline{AB}$.
(Note: we assume that there is always an ample supply of compasses and rulers of varying sizes, so that given any positive real number $r$, we can find a compass that opens wider than $r$ and a ruler that is longer than $r$).
One of the most common use of perpendicular bisectors is to find the center of a circle constructed from three points in a Euclidean plane:
Given three non collinear points $X,Y,Z$ in a Euclidean plane, let $C$ be the unique circle determined by $X,Y,Z$. Then the center of $C$ is located at the intersection of the perpendicular bisectors of $\overline{XY}$ and $\overline{YZ}$.
Title  perpendicular bisector 

Canonical name  PerpendicularBisector 
Date of creation  20130322 16:29:03 
Last modified on  20130322 16:29:03 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  18 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 51M15 
Classification  msc 51N20 
Classification  msc 51N05 
Synonym  center normal 
Related topic  Circumcircle^{} 
Defines  bisector 