Definition:Equidistance

Definition

Equidistance is one of the undefined terms in Tarski's Geometry.

Intuitively, two points $a,b$ are equidistant with respect to $c,d$ if the length of line segment $ab$ is the same as that of line segment $cd$.

However, at this point in the game, we have not yet defined line segment, or even distance.

As such, let the following ostensive definition suffice.

In the diagram, points $a,b$ are equidistant to points $c,d$, and we write $ab \equiv cd$.

However, points $a,d$ are not equidistant to points $a,b$, and we write $\neg \left({ad \equiv ab}\right)$ or $ad \not \equiv ab$.

In Euclidean 2-Space

Define the following coordinates in the $xy$-plane:

$a = \left({x_1,x_2}\right)$

$b = \left({y_1,y_2}\right)$

$c = \left({z_1,z_2}\right)$

$d = \left({u_1,u_2}\right)$

where $a,b,c,d \in \R^2$

$ab \equiv cd \dashv \vdash \left[{\left({x_1-y_1}\right)^2 + \left({x_2 - y_2}\right)^2 = \left({z_1-u_1}\right)^2 + \left({z_2-u_2}\right)^2}\right]$

Compare Distance Formula.

In Euclidean n-Space

Define the following coordinates in an Euclidean n-space:

$a = \left({x_1,x_2,\cdots,x_n}\right)$

$b = \left({y_1,y_2,\cdots,y_n}\right)$

$c = \left({z_1,z_2,\cdots,z_n}\right)$

$d = \left({w_1,w_2,\cdots,w_n}\right)$

where $a,b,c,d \in \R^n$

Then:

$ab \equiv cd \dashv \vdash \left[{\left({x_1-y_1}\right)^2 + \left({x_2 - y_2}\right)^2 + \cdots + \left({x_n - y_n}\right)^2 = \left({z_1-w_1}\right)^2 + \left({z_2 - w_2}\right)^2 + \cdots + \left({z_n - w_n}\right)^2}\right]$

Compare Vector Length.

Sources

• June 1999:  Alfred Tarski and Steven Givant: Tarski's System of Geometry (The Bulletin of Symbolic Logic Vol. 5, no. 2: 175 – 214) : Page 201
  Illustration courtesy of Steven Givant.