Neutral geometry: Let $r$, $m$ and $n$ be lines. Prove that if $r$ is parallel to $m$ and $m$ is not parallel to $n$, then $r$ is not parallel to $n$.

Question

This seems extremely obvious, but I haven't been able to prove it. I thought about doing a proof by contradiction but couldn't manage. Any help would be appreciated.

Edit: to clarify, this is neutral (plane) geometry.

Answer 1

This is equivalent to Playfair's axiom: given a line $\ell$ and a point $P$ not on $\ell$, there is at most one line through $\ell$ parallel to $P$.

To see this, note that the transitivity you want fails - $r$ is parallel to both $m$ and $n$, but $m$ is not parallel to $n$ - exactly when there are multiple parallel lines to $r$ through the intersection point of $m$ and $n$.

Thus you are out of luck in neutral geometry.