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Mathematics LibreTexts

12.8: Axiom h-V

  • Page ID
    58345
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    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 \(\ell\).

    截屏2021-02-24 下午1.33.10.png

    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.

    截屏2021-02-24 下午1.34.19.png