# Axiom:Identity of Equidistance

## Contents

## Axiom

Let $\equiv$ be the relation of equidistance.

Let $=$ be the relation of equality.

Then the following axiom is imposed:

- $\forall a, b, c: ab \equiv cc \implies a = b$

where $a, b, c$ are points.

## Intuition

If two points have no distance between them, they are the same point.

## Also see

- Identity of Points, an analogue of Identity of Equidistance in the context of Second-Order Logic.

## 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 177 : Axiom $3$