>>12954521First of all, define a unique line as we are given the axioms of a point and a line but not of a "unique" line.
Here we use axioms from ZFC, first order logic and the axioms of the Euclidean geometry.
It is possible to redefine the first axioms
1. A straight line can be drawn between any two point except some two points.
But that contradicts the second axiom therefore it will be redefined:
2. Any terminated straight line may be extended indefinitely except if it's extension lines up between specific two points (some specific points A and B)
And also,
3. A circle may be drawn with any given point as center and any given radius except with two specific points.
But who cares it's not even helpful for any reason to make this axioms
