Random triangles in planar regions
Rendiconti del Circolo Matematico di Palermo
Geometric probability, Integral formula, Planar region, Random triangle, Sylvester four-point problem
© 2018, Springer-Verlag Italia S.r.l., part of Springer Nature. 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. 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: 227+20ln281 (Halász and Kleitman in Stud Appl Math 53:225–237, 1974; Prékopa in Period Math Hung 2:259–282, 1972). 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 Limaçons r= a+ cos t, a> 1 , and O the origin of the polar coordinates, the probability is 14-12a2(4a2+1)(2a2+1)3π2.
Ionaşcu, Eugen J., "Random triangles in planar regions" (2019). Faculty Bibliography. 2754.