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arXiv · 1612.08619

Random triangles in planar regions containing a fixed point

Eugen J. Ionascu
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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 Limaçons $r=a+\cos t$, $a>1$, and $O$ the origin of the polar coordinates, the probability is $\frac{1}{4}-\frac{12a^2(4a^2+1)}{(2a^2+1)^3π^2}$.

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Eugen J. Ionascu. 2018-02-11. Random triangles in planar regions containing a fixed point. https://arxiv.org/abs/1612.08619

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