Axiom:Equal Points are Equidistant to a Third Point

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 (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $186$ : Axiom $19$ (unnamed)