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

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.55 (Compass Constructions and Angle-to-Side Triangle Reconstruction) (SeeligNotebooks).

Folio 0349 (Page 1 of 5): Compass Construction — Vesica Piscis & Orthogonal Axes (drawings-001)

[Folio 0349]

Folio 0349 (original scan 0349.jpg) from resources/seelig/55_misc.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0349 (0349.jpg, Sheet 1 of 5):

Full-page compass and straightedge construction of a vesica piscis, perpendicular bisector, and inscribed circles.

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

Editorial & Symbolic Notes

Archival scan 0349.jpg from Fred W. Seelig’s notebook dossier on F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction (Page 1 of 5). Companion OpenDocument worksheet(s): General Reference, Incircle-Incenter. Classical Euclidean foundations: I.1, IV.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0350 (Page 2 of 5): Compass Construction — Inscribed Square & Diagonal Grid (drawings-002)

[Folio 0350]

Folio 0350 (original scan 0350.jpg) from resources/seelig/55_misc.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0350 (0350.jpg, Sheet 2 of 5):

Compass construction of a circle, orthogonal diameters, and inscribed square with auxiliary secant lines.

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

Editorial & Symbolic Notes

Archival scan 0350.jpg from Fred W. Seelig’s notebook dossier on F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction (Page 2 of 5). Companion OpenDocument worksheet(s): General Reference, Incircle-Incenter. Classical Euclidean foundations: I.1, IV.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0351 (Page 3 of 5): Reconstructing a Triangle from Angles (alpha, beta, gamma) — Assumption Sets (misc-2-001)

[Folio 0351]

Folio 0351 (original scan 0351.jpg) from resources/seelig/55_misc.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0351 (0351.jpg, Sheet 3 of 5):

Sets up the problem of determining \((a,b,c)\) and vertex coordinates \((A,B,C)\) from \((\alpha, \beta, \gamma)\) under three scale normalizations (\(c=1\), \(R=1\), or \(K=1\)).

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

Editorial & Symbolic Notes

Archival scan 0351.jpg from Fred W. Seelig’s notebook dossier on F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction (Page 3 of 5). Companion OpenDocument worksheet(s): General Reference, Incircle-Incenter. Classical Euclidean foundations: I.1, IV.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0352 (Page 4 of 5): Unit-Area Normalization K = 1 & Intouch Cotangents (misc-2-002)

[Folio 0352]

Folio 0352 (original scan 0352.jpg) from resources/seelig/55_misc.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0352 (0352.jpg, Sheet 4 of 5):

Uses \(K = rs = 1 \implies s = 1/r\) and \(s - a = r\cot(\alpha/2), s - b = r\cot(\beta/2)\) to solve for \(r, s\), and \((a,b,c)\).

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

Editorial & Symbolic Notes

Archival scan 0352.jpg from Fred W. Seelig’s notebook dossier on F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction (Page 4 of 5). Companion OpenDocument worksheet(s): General Reference, Incircle-Incenter. Classical Euclidean foundations: I.1, IV.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0353 (Page 5 of 5): Side Recovery c = 2s - (a + b) & Vertex Coordinates (misc-2-003)

[Folio 0353]

Folio 0353 (original scan 0353.jpg) from resources/seelig/55_misc.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0353 (0353.jpg, Sheet 5 of 5):

Recovers \(c = (s-a)+(s-b) = 2s-(a+b)\) and writes the resulting Cartesian coordinates of \(A, B, C\).

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

Editorial & Symbolic Notes

Archival scan 0353.jpg from Fred W. Seelig’s notebook dossier on F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction (Page 5 of 5). Companion OpenDocument worksheet(s): General Reference, Incircle-Incenter. Classical Euclidean foundations: I.1, IV.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.