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

Vesica-Square Drawings and Reconstructing (a, b, c) from Interior Angles Under Scale Normalization

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).

Interactive GEOMETOR construction for F.55 (Compass Constructions and Angle-to-Side Triangle Reconstruction). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== I = incenter(A, B, C) T_A = foot(I, line(B, C)) T_B = foot(I, line(C, A)) T_C = foot(I, line(A, B)) c_in = circle(I, T_A)

Mathematical Synthesis

Closing the 55-topic archive (SeeligNotebooks), this five-folio folder pairs two pure compass-and-straightedge drawings (drawings-001–002, showing overlapping circles, a vesica piscis, and an inscribed square with perpendicular axes) with a three-page analytic study (misc-2-001–003) on reconstructing a triangle’s side lengths \((a,b,c)\) and vertex coordinates \((A,B,C)\) from its interior angles \((\alpha, \beta, \gamma)\) under three alternative scale normalizations.

Under the unit-area normalization \(K = r s = 1\) (so \(s = 1/r\)), Seelig expresses the Conway triangle notation intouch segments via the half-angle cotangents \(x = s - a = r\cot(\alpha/2)\), \(y = s - b = r\cot(\beta/2)\), and \(z = s - c = r\cot(\gamma/2)\), recovering \(c = (s-a) + (s-b) = 2s - (a+b)\) and \(r^2 = \frac{1}{\cot(\alpha/2) + \cot(\beta/2) + \cot(\gamma/2)}\).

Key Algebraic Identities

\[s - a = r\cot\frac{\alpha}{2}, \qquad s - b = r\cot\frac{\beta}{2}, \qquad s - c = r\cot\frac{\gamma}{2}\]

\[K = r s = 1 \implies s = \frac{1}{r}, \qquad c = (s - a) + (s - b) = 2 s - (a + b)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction — Folios (Folios 0349–0353, 5 scanned manuscript pages in resources/seelig/55_misc)
Euclid’s Elements	I.1, IV.4 (EuclidElements)
GEOMETOR Elements (G-Index)	G.1 point, G.8 boundary
Encyclopedia of Triangle Centers	X(1) — Incenter, X(7) — Gergonne Point (Kimberling1998)
Seelig Spreadsheets	General Reference, Incircle-Incenter (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.14 Conway Notation, F.29 Inradius, F.49 Vesica Piscis and Acute-Obtuse Probability