Axiom:Equal Points are Equidistant to a Third Point

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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.


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