Distribution of shapes of Delaunay triangles 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.

Since no one answered here, I posted this question to mathoverflow, where Igor Rivin posted this answer, which I "accepted":

> See this paper of R. E. Miles (he has plenty of related results for points on the sphere, etc, etc, mathscinet will tell you more). The results you want are in section 9 (p. 112, and thereabouts). (the paper is: On the homogeneous planar Poisson point process, R. E. Miles, Mathematical Biosciences 6 (1970).