12.8: Axiom h-V

Finally, we need to check that the Axiom h-V holds; that is, we need to prove the following claim.

Theorem $\PageIndex{1}$

Claim For any h-line $\ell$ and any h-point $P\notin\ell$ there are at least two h-lines that pass thru $P$ and have no points of intersection with $\ell$.

Instead of Proof

Applying the main observation we can assume that $P$ is the center of the absolute.

The remaining part of the proof can be guessed from the picture

Exercise $\PageIndex{1}$

Show that in the h-plane there are 3 mutually parallel h-lines such that any pair of these three lines lies on one side of the remaining h-line.

Hint

Look at the diagram and think.