F.6 Centroid — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.6 (Centroid) (SeeligNotebooks).

Folio 0022 (Page 1 of 2): Centroid Definition & Vector Average

[Folio 0022]

Folio 0022 (original scan 0022.jpg) from resources/seelig/06_centroid.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0022 (0022.jpg, Sheet 1 of 2):

Draws \(\triangle ABC\) with congruent-tick-marked midpoints and medians concurring at \(G\), defining the centroid as the concurrency of median cevians.

\[\vec{G} = \frac{\vec{A} + \vec{B} + \vec{C}}{3}, \qquad G = (1 : 1 : 1) \text{ in homogeneous barycentrics}\]

Editorial & Symbolic Notes

Archival scan 0022.jpg from Fred W. Seelig’s notebook dossier on F.6 Centroid (Page 1 of 2). Companion OpenDocument worksheet(s): General Reference, Centroid, 5-6-7 A,B,C. Classical Euclidean foundations: I.10, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid.

Folio 0023 (Page 2 of 2): The Medial Triangle M_A M_B M_C

[Folio 0023]

Folio 0023 (original scan 0023.jpg) from resources/seelig/06_centroid.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0023 (0023.jpg, Sheet 2 of 2):

Constructs the medial triangle \(\triangle M_A M_B M_C\) inscribed in \(\triangle ABC\) together with medians meeting at \(G\).

\[m_a^2 = \frac{2 b^2 + 2 c^2 - a^2}{4}, \qquad m_b^2 = \frac{2 c^2 + 2 a^2 - b^2}{4}, \qquad m_c^2 = \frac{2 a^2 + 2 b^2 - c^2}{4}\]

Editorial & Symbolic Notes

Archival scan 0023.jpg from Fred W. Seelig’s notebook dossier on F.6 Centroid (Page 2 of 2). Companion OpenDocument worksheet(s): General Reference, Centroid, 5-6-7 A,B,C. Classical Euclidean foundations: I.10, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid.