A Note on Random Triangles

In this paper we prove that the point in a triangle T maximizing the probability that it lies within a randomly chosen triangle within T is the center of T . A simple general closed expression for the indicated probability at any point P of T is obtained, though the proof proceeds by a symmetry argument on contributions to derivatives of the probability. It is also shown that the indicated probability is linearly related to the average over points in T of the product of the areas within T on either side of the line determined by P and . This gives the proof for n = 3 of the following conjecture of A. Prékopa: Fix a point P in an n − 1 dimensional simplex S . Take n independent points, uniformly distributed in S , and consider the probability ρ (P) of P being inside the random simplex determined by these n points. Conjecture: ρ (P) is maximal, when P is the center of gravity of S .