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 $\overline{AB}$.

Figure 1: The construction of a perpendicular bisector

(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	2013-03-22 16:29:03
Last modified on	2013-03-22 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