Geometric probabilities concerning large random triangles in the hyperbolic plane

Abstract

Concerning a random triangle on a disk D_R of radius R in the hyperbolic plane, the following four geometric probabilities are studied: (i) the probability p_a(R) that a random triangle is acute; (ii) the probability p_o(R) that a random triangle has the orthocenter; (iii) the probability p_e(R) that a random triangle has at least one of the three excenters; and (iv) the probability p_c(R) that a random triangle has the circumcenter. It is shown that, as R tends to the infinity, both the probability p_a(R) and p_o(R) tend to one, whereas the probability p_e(R) tends to zero. Moreover it is shown that the probability p_c(R) tends to a limit p_c, which can be expressed as a certain expectation concerning a random triangle in the Euclidean plane. To evaluate this expectation numerically, we obtain 0.45962039 as an estimate for p_c.