F.14 Conway Notation — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.14 (Conway Notation) (SeeligNotebooks).

Folio 0092 (Page 1 of 6): Conway Notation Definitions, Trig Functions & Identities

[Folio 0092]

Folio 0092 (original scan 0092.jpg) from resources/seelig/14_conway-notation.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0092 (0092.jpg, Sheet 1 of 6):

Defines \(S = 2K, S_\omega = S\cot\omega, S_A, S_B, S_C\) and records the fundamental quadratic and cubic Conway identities.

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

Editorial & Symbolic Notes

Archival scan 0092.jpg from Fred W. Seelig’s notebook dossier on F.14 Conway Notation (Page 1 of 6). Companion OpenDocument worksheet(s): General Reference, Conway's Conversions, 5-6-7 A,B,C, Barycentric. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter, X(6) — Symmedian (Lemoine) Point.

Folio 0093 (Page 2 of 6): Worked Evaluation on the 5-6-7 Reference Triangle (Part 1)

[Folio 0093]

Folio 0093 (original scan 0093.jpg) from resources/seelig/14_conway-notation.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0093 (0093.jpg, Sheet 2 of 6):

Computes \(K = 6\sqrt{6}, S = 12\sqrt{6}, S_\omega = 55, S_A = 30, S_B = 19, S_C = 6\), and vertex coordinates \((I, J) = (30/7, 12\sqrt{6}/7)\).

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

Editorial & Symbolic Notes

Archival scan 0093.jpg from Fred W. Seelig’s notebook dossier on F.14 Conway Notation (Page 2 of 6). Companion OpenDocument worksheet(s): General Reference, Conway's Conversions, 5-6-7 A,B,C, Barycentric. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter, X(6) — Symmedian (Lemoine) Point.

Folio 0094 (Page 3 of 6): Intercept N and Altitude Splits on the 5-6-7 Triangle (Part 2)

[Folio 0094]

Folio 0094 (original scan 0094.jpg) from resources/seelig/14_conway-notation.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0094 (0094.jpg, Sheet 3 of 6):

Finds the \(y\)-intercept \(N = \frac{84}{19}\sqrt{6} \approx 10.829\) and altitude base fractions \(k_c = 30/49, k_c' = 19/49\).

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

Editorial & Symbolic Notes

Archival scan 0094.jpg from Fred W. Seelig’s notebook dossier on F.14 Conway Notation (Page 3 of 6). Companion OpenDocument worksheet(s): General Reference, Conway's Conversions, 5-6-7 A,B,C, Barycentric. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter, X(6) — Symmedian (Lemoine) Point.

Folio 0095 (Page 4 of 6): Stewart’s Theorem for Altitudes in Conway Notation

[Folio 0095]

Folio 0095 (original scan 0095.jpg) from resources/seelig/14_conway-notation.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0095 (0095.jpg, Sheet 4 of 6):

Substitutes \(h_c = 2K/c\) into \(a^2 k_c + b^2 k_c' - c^2 k_c k_c' = h_c^2\) to solve for \(k_c\).

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

Editorial & Symbolic Notes

Archival scan 0095.jpg from Fred W. Seelig’s notebook dossier on F.14 Conway Notation (Page 4 of 6). Companion OpenDocument worksheet(s): General Reference, Conway's Conversions, 5-6-7 A,B,C, Barycentric. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter, X(6) — Symmedian (Lemoine) Point.

Folio 0096 (Page 5 of 6): Alternative Sum-of-Squares Notation (S = a^2 + b^2 + c^2)

[Folio 0096]

Folio 0096 (original scan 0096.jpg) from resources/seelig/14_conway-notation.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0096 (0096.jpg, Sheet 5 of 6):

Rewrites the cosine law and altitude quadratic using \(S \triangleq a^2+b^2+c^2\) and \(P \triangleq abc\).

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

Editorial & Symbolic Notes

Archival scan 0096.jpg from Fred W. Seelig’s notebook dossier on F.14 Conway Notation (Page 5 of 6). Companion OpenDocument worksheet(s): General Reference, Conway's Conversions, 5-6-7 A,B,C, Barycentric. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter, X(6) — Symmedian (Lemoine) Point.

Folio 0097 (Page 6 of 6): Clean Reference Card for Conway Symbols

[Folio 0097]

Folio 0097 (original scan 0097.jpg) from resources/seelig/14_conway-notation.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0097 (0097.jpg, Sheet 6 of 6):

Summarizes \(S_A, S_B, S_C\) as offsets from \((a^2+b^2+c^2)/2\).

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

Editorial & Symbolic Notes

Archival scan 0097.jpg from Fred W. Seelig’s notebook dossier on F.14 Conway Notation (Page 6 of 6). Companion OpenDocument worksheet(s): General Reference, Conway's Conversions, 5-6-7 A,B,C, Barycentric. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter, X(6) — Symmedian (Lemoine) Point.