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