Transitive parallel lines in noneuclidean-geometry

Question

Is it true in neutral geometry that "If a line $m$ parallel to to a line $\ell$ , and line $\ell$ parallel to line $n$ then $m$ parallel to line $n$"? ' where $m\ne n$

I think that this is corrent, it's true Poincaré disk model and also at euclidean plane

$$n------------\\ \ell------------\\ m------------$$

Answer 1

No, it is not correct. If it were correct in neutral geometry, it would be correct in every extension of neutral geometry - including hyperbolic geometry. But the standard picture of two lines through a given point - call them $m$ and $n$ - both parallel to a third line - call it $l$ - is a counterexample.

And so I think you mis-analyzed - it cannot be true in the Poincare disk model.

Answer 2

Here is a counter example in Poincaré disk model.