In Hyperbolic Geometry two points at infinity $\Omega_1$ and $\Omega_2$ have at most one line connecting them

Question

In Hyperbolic Geometry, any two points at infinity $\Omega_1$ and $\Omega_2$ have at most one line connecting them.

I have been thinking about this for a while and I believe it would be easier to do by way of contradiction but I cannot think of a way to start.

Could anyone give me a hint on how to proceed?

Thanks!

Answer 1

The short answer is that it is an axiom.

two lines can share (intersect) at most one point (ideal or not)

But then especially for ideal points.

Two lines going to the same ideal point make an angle of 0 degrees.

suppose we have at the ideal point a third line orthogonal to one of them at the ideal point, this line is also orthogonal to the other line.

But the same applies to the ideal point at the other end and that means we have a rectangle and in hyperbolic geometry rectangles do not exist.

So lines cannot share both ideal points.