F.18 Equilateral Triangle — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.18 (Equilateral Triangle) (SeeligNotebooks).

Folio 0122 (Page 1 of 3): Equilateral Triangle Fundamental Properties

[Folio 0122]

Folio 0122 (original scan 0122.jpg) from resources/seelig/18_equilateral-triangle.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0122 (0122.jpg, Sheet 1 of 3):

Lists coincidence of orthocenter, centroid, and incenter, equal altitudes, and \(h_A = 3r\) (\(h_A = \sqrt{3}/2\) for \(a=1\)).

\[h_A = \frac{\sqrt{3}}{2} a = 3 r, \qquad R = \frac{a}{\sqrt{3}} = 2 r, \qquad K = \frac{\sqrt{3}}{4} a^2\]

Editorial & Symbolic Notes

Archival scan 0122.jpg from Fred W. Seelig’s notebook dossier on F.18 Equilateral Triangle (Page 1 of 3). Companion OpenDocument worksheet(s): General Reference, Polygon Table. Classical Euclidean foundations: I.1, IV.2, IV.3. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0123 (Page 2 of 3): Incenter-Bisector Equilateral Triangle Proof (Page 1)

[Folio 0123]

Folio 0123 (original scan 0123.jpg) from resources/seelig/18_equilateral-triangle.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0123 (0123.jpg, Sheet 2 of 3):

Simplifies \(A'I = r\csc(\alpha/2) - r\cot(\alpha/2) + \frac{r}{\sqrt{3}}\cos(\alpha/2) = \frac{2r}{\sqrt{3}}\sin(\frac{\alpha}{2} + \frac{\pi}{6})\).

\[A'I = \frac{2 r}{\sqrt{3}}\sin\left(\frac{\alpha}{2} + \frac{\pi}{6}\right), \qquad P(\epsilon) = m + n\epsilon + p\epsilon^2 = 0 \quad (\epsilon = e^{i 2\pi/3})\]

Editorial & Symbolic Notes

Archival scan 0123.jpg from Fred W. Seelig’s notebook dossier on F.18 Equilateral Triangle (Page 2 of 3). Companion OpenDocument worksheet(s): General Reference, Polygon Table. Classical Euclidean foundations: I.1, IV.2, IV.3. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0124 (Page 3 of 3): Incenter-Bisector Equilateral Triangle Proof via Roots of Unity (Page 2)

[Folio 0124]

Folio 0124 (original scan 0124.jpg) from resources/seelig/18_equilateral-triangle.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0124 (0124.jpg, Sheet 3 of 3):

Substitutes \(u = \alpha/2 + \pi/6, v = \beta/2 + \pi/6, w = \gamma/2 + \pi/6\) (\(u+v+w=\pi\)) and complex cube root of unity \(\epsilon = e^{i 2\pi/3}\).

\[h_A = \frac{\sqrt{3}}{2} a = 3 r, \qquad R = \frac{a}{\sqrt{3}} = 2 r, \qquad K = \frac{\sqrt{3}}{4} a^2\]

Editorial & Symbolic Notes

Archival scan 0124.jpg from Fred W. Seelig’s notebook dossier on F.18 Equilateral Triangle (Page 3 of 3). Companion OpenDocument worksheet(s): General Reference, Polygon Table. Classical Euclidean foundations: I.1, IV.2, IV.3. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter.