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. 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: \(\frac{2}{27}+20\frac{\ln 2}{81}\) (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 \(\frac{1}{4}-\frac{12a^2(4a^2+1)}{(2a^2+1)^3\pi ^2}\).
Similar content being viewed by others
References
Baddeley, A.: A fourth note on recent research in geometrical probability. Adv. Appl. Prob. 9, 824–860 (1977)
Eisenberg, B., Sullivan, R.: Random triangles in \(n\) dimensions. Am. Math. Mon. 103(4), 308–318 (1996)
Eugen, J.: Ionascu and Gabriel Prajitura. Things to do with a broken stick. Int. J. Geom. 2(2), 5–30 (2013)
Halász, S., Kleitman, D.J.: A note on random triangles. Stud. Appl. Math. 53, 225–237 (1974)
Hall, G.R.: Acute triangles in the \(n\)-ball. J. Appl. Prob. 19, 712–715 (1982)
Henze, N.: Random triangles in convex regions. J. Appl. Prob. 20, 111–125 (1983)
Kendall, M.G., Moran, P.A.P.: Geometrical Probability. Hafner, New York (1963)
Langford, E.: The probability that a random triangle is obtuse. Biometrika 56(3), 689–690 (1969)
Prékopa, A.: On the vertices of random convex polyhedra. Period. Math. Hung. 2, 259–282 (1972)
Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)
Wagne, U., Welzl, E.: A continuous analogue of the upper bound theorem. Discrete Comput. Geom. 26, 205–219 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ionaşcu, E.J. Random triangles in planar regions. Rend. Circ. Mat. Palermo, II. Ser 68, 363–383 (2019). https://doi.org/10.1007/s12215-018-0364-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-018-0364-8