Posts tagged 'monte-carlo'

F.41 Geometric Problems and Extremal Loci

Among all triangles with longest side c = |AB| = 1, the acute locus inside the vesica piscis above the semicircle on AB has area A = int_0^{sqrt(3)} (sqrt(4 - x^2) - 1) dx - pi/4 = (5 pi - 6 sqrt(3))/12 approx 0.44297 (for c = 2), and maximizing d_IO over acute triangles approaches the degenerate right-isosceles limit C -> (0, 1).

F.49 Vesica Piscis and Acute-Obtuse Probability

When segment AB of length c is the longest side of triangle ABC, vertex C lies inside the vesica piscis of area c^2 (4 pi - 3 sqrt(3))/6; the semicircle of diameter AB partitions each quadrant of area c^2 (4 pi - 3 sqrt(3))/24 into an obtuse region (O) = c^2 pi / 16 and an acute region (A) = c^2 (5 pi - 6 sqrt(3))/48, giving an exact obtuse-to-acute probability ratio (O)/(A) = 3 pi / (5 pi - 6 sqrt(3)) approx 1.773.