F.46 Stewart’s Theorem — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.46 (Stewart’s Theorem) (SeeligNotebooks).

Folio 0293 (Page 1 of 2): Stewart’s Theorem Proof (stewart-001, FWS 11/18/2013)

[Folio 0293]

Folio 0293 (original scan 0293.jpg) from resources/seelig/46_stewarts-theorem.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0293 (0293.jpg, Sheet 1 of 2):

Defines dimensionless base fractions \(k, k'\) (\(k+k'=1\)) for cevian \(f = |CD|\) and eliminates \(\cos\theta\) between \(\triangle ACD\) and \(\triangle BCD\) to prove \(f^2 = k a^2 + k' b^2 - k k' c^2\).

\[f^2 = k a^2 + k' b^2 - k k' c^2 \qquad (k + k' = 1,\; |AD| = k' c,\; |DB| = k c)\]

Editorial & Symbolic Notes

Archival scan 0293.jpg from Fred W. Seelig’s notebook dossier on F.46 Stewart’s Theorem (Page 1 of 2). Companion OpenDocument worksheet(s): Stewart's Theorem, Ceva & Menelaus, Centroid. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid.

Folio 0294 (Page 2 of 2): Apollonius’s Median Formula as Special Case (stewart-002)

[Folio 0294]

Folio 0294 (original scan 0294.jpg) from resources/seelig/46_stewarts-theorem.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0294 (0294.jpg, Sheet 2 of 2):

Evaluates Stewart’s identity at the midpoint \(k = k' = 1/2\) to obtain \(f^2 = \frac{1}{2}(a^2 + b^2 - \frac{1}{2}c^2)\).

\[k = k' = \frac{1}{2} \implies m_c^2 = \frac{1}{2}\left(a^2 + b^2 - \frac{1}{2}c^2\right) = \frac{2 a^2 + 2 b^2 - c^2}{4}\]

Editorial & Symbolic Notes

Archival scan 0294.jpg from Fred W. Seelig’s notebook dossier on F.46 Stewart’s Theorem (Page 2 of 2). Companion OpenDocument worksheet(s): Stewart's Theorem, Ceva & Menelaus, Centroid. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid.