F.29 Inradius — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.29 (Inradius) (SeeligNotebooks).

Folio 0207 (Page 1 of 3): Inradius & Special Inradius Values

[Folio 0207]

Folio 0207 (original scan 0207.jpg) from resources/seelig/29_inradius.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0207 (0207.jpg, Sheet 1 of 3):

Records algebraic and trigonometric formulas for \(r\) alongside a special-value table for Pythagorean triples \((3,4,5), (5,12,13), (8,15,17), (7,24,25)\), the \(30^\circ\text{--}60^\circ\text{--}90^\circ\) triangle, and the golden triangle \((\phi b, \phi b, b)\) with \(r = \frac{b}{2}\sqrt{5-2\sqrt{5}}\).

\[r = \frac{K}{s} = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(a+b-c)}{a+b+c}} = \sqrt{\frac{x y z}{x + y + z}}\]

Editorial & Symbolic Notes

Archival scan 0207.jpg from Fred W. Seelig’s notebook dossier on F.29 Inradius (Page 1 of 3). Companion OpenDocument worksheet(s): Incircle-Incenter, General Reference. Classical Euclidean foundations: IV.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0208 (Page 2 of 3): Conway Intouch Segments & Gergonne Point (Page 1)

[Folio 0208]

Folio 0208 (original scan 0208.jpg) from resources/seelig/29_inradius.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0208 (0208.jpg, Sheet 2 of 3):

Relates \(x = s-a, y = s-b, z = s-c\) to the incircle contact points \(T_A, T_B, T_C\) and the Gergonne point \(G_e = AT_A \cap BT_B \cap CT_C\), deriving \(r^2 = xyz/s\).

\[r = 4 R \sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}, \qquad x = s - a,\; y = s - b,\; z = s - c\]

Editorial & Symbolic Notes

Archival scan 0208.jpg from Fred W. Seelig’s notebook dossier on F.29 Inradius (Page 2 of 3). Companion OpenDocument worksheet(s): Incircle-Incenter, General Reference. Classical Euclidean foundations: IV.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0209 (Page 3 of 3): Measured Scalene Verification (Page 2)

[Folio 0209]

Folio 0209 (original scan 0209.jpg) from resources/seelig/29_inradius.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0209 (0209.jpg, Sheet 3 of 3):

Constructs the incircle and Gergonne cevians on a measured triangle (\(a=28.3, b=19, c=30.8, s=39, x=10.7, y=20, z=8.2, K=263\)) and verifies \(r = K/s \approx 6.74\).

\[r = \frac{K}{s} = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(a+b-c)}{a+b+c}} = \sqrt{\frac{x y z}{x + y + z}}\]

Editorial & Symbolic Notes

Archival scan 0209.jpg from Fred W. Seelig’s notebook dossier on F.29 Inradius (Page 3 of 3). Companion OpenDocument worksheet(s): Incircle-Incenter, General Reference. Classical Euclidean foundations: IV.4. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.