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?
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.