I'm studying triangulations of the hyperbolic plane and have come across the following theorem:
If we are given a triangle $\Delta_0$ with angles $\pi$/l,$\pi$/m,$\pi$/n, where the integers l, m, n satisfy $1/l+1/m+1/n<1$, if we obtain further triangles by reflection in the sides of $\Delta_0$, then obtain further triangles by reflections in the sides of the new ones, and so on, the result is a set of infinitely many triangles, no two of which overlap (adjacent ones have common boundary points), and which together cover the entire hyperbolic plane. The entire structure is invariant with respect to each of the reflections mentioned.
There is no proof given in the text book I'm reading (although a proof of one case is given), whilst it does provide a source for the claim, I can't find the book it references. If somebody could provide a proof or point me towards a source for a proof, that would be great.
This is a special case of the Poincare polygon theorem. Typing that phrase into Google gives several hits to different people's lecture notes. You can also look it up in Ratcliffe's book Foundations of hyperbolic manifolds.