How to show the Poincaré disk is hyperbolic for some $\delta$ I am trying to prove that the Poincaré disk, $\mathbb{D}$, is $\delta$-hyperbolic with respect to the slim triangle definition for hyperbolicity. I have been stuck for a while on where to begin, particularly with respect to ideal triangles. 
I am trying to do this from a Geometrical Group Theory point of view, as I am not familiar enough with topology to use those tools. 
Many thanks

The orientation-preserving isometry group is $PSL(2,R)$ and it acts transitively on triples of pairwise distinct points on the ideal boundary, i.e. on the set of non-degenerate ideal triangles. In other words: all ideal triangles are isometric. Thus it suffices to compute $\delta$ for one ideal triangle. You will get $\delta=\log(\sqrt{2}+1)$.
Then you still have to prove that this implies the same estimate for non-ideal triangles.
