I'm trying to prove that, in the Poincaré half-space of dimension 2, a hyperbolic ball with center $P:=(x,y)$ and radius $r$ is exactly, as a set of points, a euclidean ball with center $(x,y\cosh(r))$ and radius $r_1:=y\sinh(r)$. The distance function I'm using in the Poincaré half-plane is, written in complex coordinates, $$d_h(z,w)=2\log\frac{|z-w|+|z-\bar w|}{2\sqrt{\mathrm{Im}(z)\mathrm{Im}(w)}}.$$
My idea is to consider a generic point on the euclidean ball boundary cited above, in the form $$A:=(r_1\cos(\theta),r_1\sin(\theta)+y\cosh(r))$$ and then to explicitly compute its distance to the center of the hyperbolic ball with which we started, to prove it is exactly $r$, i.e. I'm trying to prove that $$d_h(A,P)=r.$$ This leads me towards extremely hard computations in which I have to admit I've been losing myself several times. I found the statement of the result stated on the wikipedia page https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model, but I can't find a reference for a proof. Could anyone please share some ideas or references on how to prove this result?
