Axiom:Identity of Equidistance

Axiom
Let $\equiv$ be the relation of equidistance.

Let $=$ be the relation of equality.

Then the following axiom is imposed:
 * $\forall a, b, c: ab \equiv cc \implies a = b$

where $a, b, c$ are points.

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