Posts tagged 'vesica-piscis'
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.
F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction
Given the three interior angles (alpha, beta, gamma) of a triangle normalized to unit area K = 1 (so s = 1/r), the Conway intouch segments x = s - a = r cot(alpha/2), y = s - b = r cot(beta/2), and z = s - c = r cot(gamma/2) determine the side lengths a = y + z, b = z + x, and c = x + y = 2 s - (a + b).