F.14 Conway Notation

John H. Conway’s Algebraic Triangle Symbols, Brocard Angle, and Altitude Partitions

In John H. Conway’s triangle notation, setting S = 2K (twice the area of triangle ABC), S_omega = (a^2 + b^2 + c^2)/2, and S_A = (b^2 + c^2 - a^2)/2, S_B = (c^2 + a^2 - b^2)/2, S_C = (a^2 + b^2 - c^2)/2 reduces trigonometric and barycentric identities to polynomial expressions satisfying S_B S_C + S_C S_A + S_A S_B = S^2.

Interactive GEOMETOR construction for F.14 (Conway Notation). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== H = orthocenter(A, B, C) O = circumcenter(A, B, C) G = centroid(A, B, C) Euler = line(O, H)

Mathematical Synthesis

John H. Conway’s algebraic notation (Conway triangle notation) is the primary symbolic language used across Seelig’s notebooks and Paul Yiu’s Introduction to the Geometry of the Triangle (Yiu2001, SeeligNotebooks).

In Folios 1–2, Seelig evaluates all Conway invariants on his canonical reference triangle \((a, b, c) = (5, 6, 7)\): \(K = 6\sqrt{6}\), \(S = 12\sqrt{6}\), \(S_\omega = 55\), \(S_A = 30\), \(S_B = 19\), \(S_C = 6\), altitude \(J = h_c = \frac{12\sqrt{6}}{7}\), projection \(I = \frac{30}{7}\), \(y\)-intercept \(N = \frac{84}{19}\sqrt{6}\), and fractional base splits \(k_c = \frac{30}{49}\) and \(k_c' = \frac{19}{49}\).

Key Algebraic Identities

\[S = 2 K, \qquad S_\omega = \frac{a^2 + b^2 + c^2}{2} = S \cot\omega, \qquad S_A = \frac{b^2 + c^2 - a^2}{2} = b c \cos\alpha\]

\[S^2 = b^2 c^2 - S_A^2 = S_A S_B + S_B S_C + S_C S_A, \qquad S_A S_B S_C = S^2 (S_\omega - 4 R^2)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.14 Conway Notation — Folios (Folios 0092–0097, 6 scanned manuscript pages in resources/seelig/14_conway-notation)
Euclid’s Elements	II.12, II.13 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter, X(6) — Symmedian (Lemoine) Point (Kimberling1998)
Seelig Spreadsheets	General Reference, Conway's Conversions, 5-6-7 A,B,C, Barycentric (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.3 Biquadratic Forms in Heron’s Formula

Required for: F.21 Fermat Point, F.29 Inradius, F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety, F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem, F.52 Homogeneous Barycentric Coordinates, F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction