The theorem I want to prove is this
Given a geodesic $I$ in hyperbolic space and a point $p$ not on $I$, there is a unique point $q$ on $I$ such that the geodesic joining $p$ and $q$ meets $I$ orthogonally.
I'm working inside $D$ (Poincaré disk model). I already received some advice, which is to consider some arbitrary geodesic while taking $p$ to be at the origin. I've drawn a picture which shows this idea; the geodesic $I$ is the semi-circle, and $p$ is the point at the origin not on $I$. The geodesic straight line through the origin joins $p$ and $q$, and clearly meets $I$ orthogonally.
My question is, how can I generalise this result to conclude that it's true for all points $p$ outside the geodesic?
As far as I understand, Möbius transformations map geodesics to geodesics and also act transitively on pairs (and triples) of points. I also know that Möbius transformations preserve angles, so if I map these two geodesics to two others the orthogonality will be preserved. I think I should use this information but don't how know to put it together.