Axiom:Equal Points are Equidistant to a Third Point

Axiom
Let $a$ and $b$ be points.

Let $\equiv$ be the relation of equidistance.

Let $=$ be the relation of equality.

Then the following axiom is imposed:


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

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