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Is the hyperbolic version of Sylvester co linear problem true?

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    $\begingroup$ Use the Klein model-- geodesics are straight lines. $\endgroup$
    – Neal
    Commented Sep 26, 2016 at 14:45
  • $\begingroup$ Kelly's proof on your link works in absolute geometry. $\endgroup$
    – Fedor Petrov
    Commented Sep 26, 2016 at 16:58

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Yes. Use the projective (Cayley-Klein) model of the hyperbolic plane. Your points lie inside a disk. Hyperbolic lines are chords of that disk. Since the Euclidean Silvester-Gallai is true, you have the same conclusion in the hyperbolic case.

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  • $\begingroup$ You should say "Cayley–Klein discovered by Beltrami" $\endgroup$
    – Anton Petrunin
    Commented Sep 29, 2016 at 17:19
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    $\begingroup$ Hi @AntonPetrunin Nothing against Beltrami, but then I will better say "projective model". $\endgroup$
    – Ivan Izmestiev
    Commented Sep 29, 2016 at 17:51

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