Random triangles in planar regions containing a fixed point

TL;DR

This paper derives exact formulas for the probability that a random triangle within a planar region contains a fixed point, with applications to classical geometric probability problems and specific region families.

Contribution

It introduces new integral formulae for geometric probability, extending known results and applying them to various planar regions including polygons and limaçons.

Findings

Exact probability formulas for various regions including equilateral triangles and regular polygons.

Derived a stability result for the probability of containing a fixed point.

Computed probabilities for limaçons and other families of regions.

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 $\frac{1}{4}-\frac{12a^2(4a^2+1)}{(2a^2+1)^3\pi^2}$.