Perpendicular bisector

Builds on Midpoint Vectors are perpendicular if and only if their dot product is zero Euclidean distance

Definition

Let $A$ and $B$ be distinct points of the Euclidean space $\mathbb{R}^n$ , and let $M = \frac{1}{2}(A + B)$ be their midpoint. Let $\cdot$ be the dot product. The perpendicular bisector of $A$ and $B$ is the set $\{\, X \in \mathbb{R}^n \;:\; (X - M) \cdot (B - A) = 0 \,\}.$

Remarks

A point $X$ is on the perpendicular bisector if and only if $d(X, A) = d(X, B)$ , where $d$ is the Euclidean distance. It is a hyperplane with the normal vector $B - A$ . In the plane it is a line, and in $\mathbb{R}^3$ it is a plane. Mathlib's AffineSubspace.perpBisector also allows $A = B$ . Then the set is the whole space. The perpendicular bisectors of the three sides of a triangle in the plane meet at the circumcentre.

Reading path

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Angle between two vectors concept
Euclidean distance concept
Length of a vector concept
Midpoint concept
Geometric formula for the dot product statement
Vectors are perpendicular if and only if their dot product is zero statement
Perpendicular bisector definition

Tag	`03J4NZFYRX`
URI	`m2://03J4NZFYRX`
Kind	definition
Writing	W0 stub
MSC	51M04
Defines	Perpendicular bisector
Uses	Midpoint, Vectors are perpendicular if and only if their dot product is zero, Euclidean distance

Perpendicular bisector

Definition

Remarks

Reading path

Further reading

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