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Given two points $p,q$ in the hyperbolic plane, show that the set of points equidistant from $p$ and $q$ is a hyperbolic line.

I am unsure how to proceed with this question. Would it be easier to use the upper-half plane model of hyperbolic space, or the Poincare disk model? Also, would a geometric construction help, or do I have to proceed in a more rigorous way?

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  • $\begingroup$ Depends entirely on your approach, i.e. order of your book, because it is true. So, do you know the isometries in the upper half plane model? $\endgroup$ – Will Jagy Mar 4 '13 at 23:23
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    $\begingroup$ Geometric constructions are perfectly rigorous. You don't have to do anything you don't need to. $\endgroup$ – Loki Clock Mar 4 '13 at 23:26
  • $\begingroup$ In the upper half plane model isometries come from the Möbius group I think $\endgroup$ – user55225 Mar 4 '13 at 23:28
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    $\begingroup$ Yes, that plus a single orientation-reversing reflection if desired, across a vertical line. Anyway, move so that your two points have the same imaginary coordinate. Then the perpendicular bisector of the ordinary segment between them is a geodesic. Now move back to original. Or, as Loki says, do the whole thing intrinsically. $\endgroup$ – Will Jagy Mar 4 '13 at 23:31
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Hint 1: Find a circle inversion that takes $p$ to $q$. This is an isometry, so what can you conclude about its fixed-point set? (extra hint: consult Will Jagy's comment)

Hint 2: Write down a Möbius transformation ($\operatorname{SL}_2(\mathbb{R})$ for the upper-half plane or $\operatorname{SU}(1,1)$ for the Poincaré disk) that takes $p$ to $q$. Identify its fixed-point set. Argue as in 1.

For geometric constructions (i.e., playing with circles), I find the Poincaré disk much more intuitive. Your mileage may vary.

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