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.