Axiom:Identity of Equidistance

Axiom
Let $a,b,c$ be points.

Let $\equiv$ be the relation of equidistance.

Let $=$ be the relation of equality.

Then the following axiom is imposed:


 * $\forall a,b,c: ab \equiv cc \implies a = b$

Intuition
If two points have no distance between them, they are the same point.