F.38 Pedal Triangles — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.38 (Pedal Triangles) (SeeligNotebooks).

Folio 0236 (Page 1 of 2): Pedal Triangle Side Lengths & Area Formula

[Folio 0236]

Folio 0236 (original scan 0236.jpg) from resources/seelig/38_pedal-triangles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0236 (0236.jpg, Sheet 1 of 2):

Records the trilinear pedal side-length formula for \(a', b', c'\) and Euler’s pedal area relation \((\Delta P) = \frac{R^2-OP^2}{4R^2}\Delta\).

\[a' = \frac{a b c \sqrt{\tau_B^2 + \tau_C^2 + 2\tau_B\tau_C\cos\alpha}}{2 R (a\tau_A + b\tau_B + c\tau_C)}, \qquad \Delta(P) = \frac{R^2 - OP^2}{4 R^2}\Delta\]

Editorial & Symbolic Notes

Archival scan 0236.jpg from Fred W. Seelig’s notebook dossier on F.38 Pedal Triangles (Page 1 of 2). Companion OpenDocument worksheet(s): Pedal Triangle, Orthic Triangle. Classical Euclidean foundations: I.12, I.47. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0237 (Page 2 of 2): Carnot’s Alternating Square Law & Trilinear Matrix

[Folio 0237]

Folio 0237 (original scan 0237.jpg) from resources/seelig/38_pedal-triangles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0237 (0237.jpg, Sheet 2 of 2):

Writes the alternating segment square equality \({AC'}^2 + {BA'}^2 + {CB'}^2 = {C'B}^2 + {A'C}^2 + {B'A}^2\) (‘wow!’) and the \(3\times 3\) trilinear coordinate matrix of \(A', B', C'\).

\[{AC'}^2 + {BA'}^2 + {CB'}^2 = {C'B}^2 + {A'C}^2 + {B'A}^2\]

Editorial & Symbolic Notes

Archival scan 0237.jpg from Fred W. Seelig’s notebook dossier on F.38 Pedal Triangles (Page 2 of 2). Companion OpenDocument worksheet(s): Pedal Triangle, Orthic Triangle. Classical Euclidean foundations: I.12, I.47. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.