Robert Hill wrote: > [ snip ] > More seriously, there is an assumption here > that the answer for a square is the same as "the answer" (whatever that means) > for the whole plane. I have an informal argument (which I suspect > I could make rigorous) that the probability of obtuseness for triangles whose > vertices are uniformly distributed on any convex set of finite area is < 3/4, > but I still suspect that the best answer for "a uniform distribution on > the whole plane" is = 3/4. But that's like a statement about unicorns. >
Your argument for any convex set cannot be correct. Consider a very, very long and very, very thin rectangle. I claim that if the vertices of a triangle are uniformly distributed in such a set, then the probability of obtuseness approaches 1 -- and I invite you to consider why this must (or might) be so.