F.41 Geometric Problems and Extremal Loci

Maximizing Euler Distance d_IO for Acute Triangles, Monte Carlo Vesica Integration, and 25-Problem Compendium

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).

Interactive GEOMETOR construction for F.41 (Geometric Problems and Extremal Loci). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== M = midpoint(A, B) c_semi = circle(M, A) c_left = circle(A, B) c_right = circle(B, A) O = circumcenter(A, B, C) I = incenter(A, B, C)

Mathematical Synthesis

This 10-folio collection (SeeligNotebooks) combines two analytic geometry sheets (Folios 1–2), a dated calculus and Monte Carlo investigation Find an acute triangle that maximizes :math:`d_{IO}` (FWS 2015/04/24, Folios 3–4), and a 25-problem compendium (Problem Set, Folios 5–10) that served as the blueprint for Seelig’s notebook archive.

In his April 24, 2015 study, Seelig fixes base \(c = 2\) and constrains vertex C to the region where \(c\) is the longest side (inside the vesica piscis of radii 2 centered at A and B) while \(\triangle ABC\) remains acute (outside the semicircle csemi of radius 1 and between the vertical lines at A and B). Integrating \(A = \int_0^{\sqrt{3}}(\sqrt{4-x^2}-1)\,dx - \frac{\pi}{4} = \frac{5\pi - 6\sqrt{3}}{12} \approx 0.44297\), he explains why his spreadsheet Monte Carlo sampler on \([0,\sqrt{3}]\times[0,1]\) has an acceptance rate of \(0.44297/\sqrt{3} \approx 25.6\%\), and finds that \(d_{IO}\) increases monotonically toward the right-triangle boundary \(C \to (0, 1)\).

Key Algebraic Identities

\[A_{\text{acute}} = \int_0^{\sqrt{3}} \left(\sqrt{4 - x^2} - 1\right) dx - \frac{\pi}{4} = \frac{5\pi - 6\sqrt{3}}{12} \approx 0.44297\]

\[p_{\text{accept}} = \frac{A_{\text{acute}}}{\sqrt{3}} \approx 0.25575, \qquad d_{IO} = \sqrt{R^2 - 2 r R}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.41 Geometric Problems and Extremal Loci — Folios (Folios 0257–0266, 10 scanned manuscript pages in resources/seelig/41_problems)
Euclid’s Elements	I.1, III.31 (EuclidElements)
GEOMETOR Elements (G-Index)	G.1 point, G.7 plane surface
Encyclopedia of Triangle Centers	X(1) — Incenter, X(3) — Circumcenter (Kimberling1998)
Seelig Spreadsheets	Monte Carlo, General Reference (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.32 Markowski’s Theorem, F.35 Obtuse and Acute Triangles, F.37 Euler’s Inradius Parabola

Required for: F.49 Vesica Piscis and Acute-Obtuse Probability