Do you know any proof of the fact that $\mathbb H^n$ is Rips-hyperbolic (i.e., geodesic triangles are $\delta$-slim for some $\delta$, also called "Gromov-hyperbolic" in some contexts), which makes no use of the Klein model, the ball model or the upper halfspace model for $\mathbb H^n$, but just of the hyperboloid model (i.e.: points in $\mathbb R^{n+1}$ whose Minkowski quadratic form is $-1$, with distance $d(u,v)=arcosh(-\langle u,v\rangle)$ )?
Thank you in advance.