F.19 Equal Incircles Theorem — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.19 (Equal Incircles Theorem) (SeeligNotebooks).

Folio 0125 (Page 1 of 2): Equal Incircles Theorem Statement & Ray-Fan Diagrams

[Folio 0125]

Folio 0125 (original scan 0125.jpg) from resources/seelig/19_equal-incircle-theorem.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0125 (0125.jpg, Sheet 1 of 2):

States the theorem with two ray-fan diagrams and introduces the hyperbolic substitution \(\tan\gamma_n = \sinh(a + nb)\).

\[(h - r)(\tan\gamma_{n+1} - \tan\gamma_n) = r(\sec\gamma_n + \sec\gamma_{n+1})\]

Editorial & Symbolic Notes

Archival scan 0125.jpg from Fred W. Seelig’s notebook dossier on F.19 Equal Incircles Theorem (Page 1 of 2). Companion OpenDocument worksheet(s): Incircle-Incenter. Classical Euclidean foundations: IV.4, VI.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter.

Folio 0126 (Page 2 of 2): Similarity Proof & Hyperbolic Identity (‘interesting!!!’)

[Folio 0126]

Folio 0126 (original scan 0126.jpg) from resources/seelig/19_equal-incircle-theorem.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0126 (0126.jpg, Sheet 2 of 2):

Uses \(\triangle OWX \sim \triangle PQX\) to derive \((h-r)(\tan\gamma_{n+1}-\tan\gamma_n) = r(\sec\gamma_n+\sec\gamma_{n+1})\) and reduces it to \(\frac{r}{h-r} = \tanh(b/2)\).

\[\tan\gamma_n = \sinh(a + n b), \qquad \sec\gamma_n = \cosh(a + n b), \qquad \frac{r}{h - r} = \tanh\frac{b}{2}\]

Editorial & Symbolic Notes

Archival scan 0126.jpg from Fred W. Seelig’s notebook dossier on F.19 Equal Incircles Theorem (Page 2 of 2). Companion OpenDocument worksheet(s): Incircle-Incenter. Classical Euclidean foundations: IV.4, VI.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter.