TITLE: Distribution of shapes of Delaunay triangles
QUESTION [5 upvotes]: Does anyone know the probability distribution of the shapes of Delaunay triangles in a constant-intensity Poisson process in the plane?
Slightly later edit: One can imagine performing the experiment repeatedly and looking at the one triangle that surrounds the origin, and ask, for example, how frequently it will be obtuse; or one can imagine doing it just once and looking at all of the infinitely many triangles and asking what proportion of them are obtuse.  One would (or at least I would) initially guess the two answers are the same (and similarly for other sets of shapes besides the set of all obtuse triangles).  One complication in proving that would be that the shapes of the infinitely many triangles one gets by doing the experiment once are not mutually independent.

REPLY [2 votes]: Take a look in the relevant chapter of "Spatial Tessellations" by Okabe et al. They give explicit distribution functions for a pair of angles of a random triangle in a Poisson Delaunay Tessellation, and there's also a 'bell diagram', giving the distribution of shapes and their frequencies.
If you don't have the book, I think the distribution function also appears in the paper "The Expected Extremes in the Delaunay Triangulation" - Eppstein et al.