Random triangles in planar regions containing a fixed point

Abstract

In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region (any Lebesgue measurable set of finite measure) contains a given fixed point $O$. These formulae are in terms of one integration of an appropriate function, with respect to a density function which depends of the point $O$. The formulae provide another way to approach the Sylvester's Four-Point Problem as we show in the last section. A stability result is derived for the probability. We recover the known probability in the case of an equilateral triangle and its center of mass: $\frac{2}{27}+20\frac{\ln 2}{81}$. We compute this probability in the case of a regular polygon and its center of mass for the point $O$. Other families of regions are studied. For the family of Lima\c{c}ons $r=a+\cos t$, $a>1$, and $O$ the origin of the polar coordinates, the probability is $\frac{1}{4}-\frac{12a^2(4a^2+1)}{(2a^2+1)^3\pi^2}$.