F.49 Vesica Piscis and Acute-Obtuse Probability

Locus of Longest-Side Triangles, Semicircle Partition, and Exact Area Ratio 3 pi / (5 pi - 6 sqrt(3))

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.

Interactive GEOMETOR construction for F.49 (Vesica Piscis and Acute-Obtuse Probability). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point ===== c_left = circle(A, B) c_right = circle(B, A) C = intersection(c_left, c_right) M = midpoint(A, B) c_semi = circle(M, A)

Mathematical Synthesis

Titled Notes on the locus of points C such that c = |AB| is the longest side (SeeligNotebooks), this three-folio jewel connects Euclid’s Opening Proposition I.1 (the vesica piscis formed by cleft and cright of radius \(c\)) with III.31 (Thales’s semicircle csemi on diameter A B of radius \(c/2\)). Because \(|AC| \le c\) and \(|BC| \le c\), any triangle with longest side \(c\) has vertex C inside the vesica piscis; by fourfold symmetry, Seelig analyzes a single upper-half quadrant of total area \((O) + (A) = c^2\left(\frac{\pi}{6} - \frac{\sqrt{3}}{8}\right)\).

Inside the Thales quarter-disk of radius \(c/2\), the angle at C is obtuse (\(\gamma > 90^\circ\)), giving obtuse area \((O) = \frac{1}{4}\pi(c/2)^2 = c^2\frac{\pi}{16}\). Subtracting \((O)\) from the quarter-vesica leaves the acute region \((A) = c^2\frac{5\pi - 6\sqrt{3}}{48}\). Dividing the two yields the exact geometric probability ratio \(\frac{(O)}{(A)} = \frac{3\pi}{5\pi - 6\sqrt{3}} \approx 1.7731\) (63.94% obtuse vs. 36.06% acute)—confirming why random triangles in the Monte Carlo spreadsheet are predominantly obtuse!

Key Algebraic Identities

\[(O) + (A) = c^2\left(\frac{\pi}{6} - \frac{\sqrt{3}}{8}\right) = c^2\frac{4\pi - 3\sqrt{3}}{24}, \qquad (O) = \frac{1}{4}\pi\left(\frac{c}{2}\right)^2 = c^2\frac{\pi}{16}\]

\[(A) = c^2\frac{5\pi - 6\sqrt{3}}{48}, \qquad \frac{(O)}{(A)} = \frac{3\pi}{5\pi - 6\sqrt{3}} \approx 1.7731\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.49 Vesica Piscis and Acute-Obtuse Probability — Folios (Folios 0315–0317, 3 scanned manuscript pages in resources/seelig/49_vesica-pisces)
Euclid’s Elements	I.1, III.31 (EuclidElements)
GEOMETOR Elements (G-Index)	G.1 point, G.7 plane surface
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Monte Carlo, General Reference (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.18 Equilateral Triangle, F.35 Obtuse and Acute Triangles, F.41 Geometric Problems and Extremal Loci

Required for: F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction