F.21 Fermat Point — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.21 (Fermat Point) (SeeligNotebooks).

Folio 0128 (Page 1 of 2): Torricelli’s Construction of the Fermat Point X(13) (FWS July 6, 2011)

[Folio 0128]

Folio 0128 (original scan 0128.jpg) from resources/seelig/21_fermat-point.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0128 (0128.jpg, Sheet 1 of 2):

Step-by-step ruler-and-compass algorithm constructing external equilateral triangle \(\triangle ABX\) and its circumcircle to locate \(F = X(13)\).

\[\angle AFB = \angle BFC = \angle CFA = 120^\circ, \qquad \min_{P} (|PA| + |PB| + |PC|) = |CC'| = \sqrt{\frac{a^2 + b^2 + c^2}{2} + 2\sqrt{3} K}\]

Editorial & Symbolic Notes

Archival scan 0128.jpg from Fred W. Seelig’s notebook dossier on F.21 Fermat Point (Page 1 of 2). Companion OpenDocument worksheet(s): General Reference, Barycentric. Classical Euclidean foundations: I.1, III.21, III.22. Kimberling Encyclopedia of Triangle Centers loci: X(13) — First Fermat (Torricelli) Point, X(14) — Second Fermat Point, X(15) — First Isodynamic Point, X(16) — Second Isodynamic Point.

Folio 0129 (Page 2 of 2): Obtuse Triangle Regime & Distinction from the Steiner Point (Wells 1991, p. 75)

[Folio 0129]

Folio 0129 (original scan 0129.jpg) from resources/seelig/21_fermat-point.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0129 (0129.jpg, Sheet 2 of 2):

Explains why \(S_{\text{TOT}} = SA + SB + SC\) is minimized at the obtuse vertex whenever an angle is \(\ge 120^\circ\).

\[F = X(13) = \left(\csc\left(\alpha + \frac{\pi}{3}\right) : \csc\left(\beta + \frac{\pi}{3}\right) : \csc\left(\gamma + \frac{\pi}{3}\right)\right)\]

Editorial & Symbolic Notes

Archival scan 0129.jpg from Fred W. Seelig’s notebook dossier on F.21 Fermat Point (Page 2 of 2). Companion OpenDocument worksheet(s): General Reference, Barycentric. Classical Euclidean foundations: I.1, III.21, III.22. Kimberling Encyclopedia of Triangle Centers loci: X(13) — First Fermat (Torricelli) Point, X(14) — Second Fermat Point, X(15) — First Isodynamic Point, X(16) — Second Isodynamic Point.