# Axiom:Equal Points are Equidistant to a Third Point

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## Axiom

Let $\equiv$ be the relation of equidistance.

Let $=$ be the relation of equality.

Then the following axiom holds:

- $\forall a, b, c: a = b \implies ac \equiv bc$

where $a, b, c$ are points.

## Intuition

If two points are the same point, they are equidistant to a third point.

## 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 186 : Axiom $19$ (unnamed)