F.10 Circles and Geometric Mean — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.10 (Circles and Geometric Mean) (SeeligNotebooks).

Folio 0062 (Page 1 of 4): Equations of Circles & Geometric Mean

[Folio 0062]

Folio 0062 (original scan 0062.jpg) from resources/seelig/10_circles-and-geometric-mean.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0062 (0062.jpg, Sheet 1 of 4):

Expresses \(P^2 = (3+Q)(3-Q)\) and \(P^2 = (5+(Z-Q))(5-(Z-Q))\) as semicircle geometric means and derives \(Z(Q-P) = 12\) and \(Q = \frac{Z^2-16}{2Z}\).

\[G = |CH| = \sqrt{m n}, \qquad A_{\text{mean}} = |OC| = \frac{m + n}{2}, \qquad H_{\text{mean}} = |CK| = \frac{2 m n}{m + n}\]

Editorial & Symbolic Notes

Archival scan 0062.jpg from Fred W. Seelig’s notebook dossier on F.10 Circles and Geometric Mean (Page 1 of 4). Companion OpenDocument worksheet(s): General Reference, Harmonic Conjugates. Classical Euclidean foundations: II.14, III.31, VI.8, VI.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter, X(5) — Nine-Point Center, X(6) — Symmedian (Lemoine) Point, X(7) — Gergonne Point, X(8) — Nagel Point.

Folio 0063 (Page 2 of 4): Area Partition of Square ABCD into Four Rectangles

[Folio 0063]

Folio 0063 (original scan 0063.jpg) from resources/seelig/10_circles-and-geometric-mean.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0063 (0063.jpg, Sheet 2 of 4):

Partitions the square of area \(Z^2\) around \(P(P, Q)\) into four rectangles ①–④ and relates their dimensions to the circle equations.

\[G = |CH| = \sqrt{m n}, \qquad A_{\text{mean}} = |OC| = \frac{m + n}{2}, \qquad H_{\text{mean}} = |CK| = \frac{2 m n}{m + n}\]

Editorial & Symbolic Notes

Archival scan 0063.jpg from Fred W. Seelig’s notebook dossier on F.10 Circles and Geometric Mean (Page 2 of 4). Companion OpenDocument worksheet(s): General Reference, Harmonic Conjugates. Classical Euclidean foundations: II.14, III.31, VI.8, VI.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter, X(5) — Nine-Point Center, X(6) — Symmedian (Lemoine) Point, X(7) — Gergonne Point, X(8) — Nagel Point.

Folio 0064 (Page 3 of 4): Homogeneous Barycentric Coordinates Table for I, O, H, Ge, Na, K, N

[Folio 0064]

Folio 0064 (original scan 0064.jpg) from resources/seelig/10_circles-and-geometric-mean.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0064 (0064.jpg, Sheet 3 of 4):

Lists normalized HBC formulas in terms of \(S = a^2+b^2+c^2\) and \(x, y, z\), noting * This expression not found in ETC for \(H = (S-2b^2)(S-2c^2)/D\).

\[G = |CH| = \sqrt{m n}, \qquad A_{\text{mean}} = |OC| = \frac{m + n}{2}, \qquad H_{\text{mean}} = |CK| = \frac{2 m n}{m + n}\]

Editorial & Symbolic Notes

Archival scan 0064.jpg from Fred W. Seelig’s notebook dossier on F.10 Circles and Geometric Mean (Page 3 of 4). Companion OpenDocument worksheet(s): General Reference, Harmonic Conjugates. Classical Euclidean foundations: II.14, III.31, VI.8, VI.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter, X(5) — Nine-Point Center, X(6) — Symmedian (Lemoine) Point, X(7) — Gergonne Point, X(8) — Nagel Point.

Folio 0065 (Page 4 of 4): Quadratic Maximization Exercise

[Folio 0065]

Folio 0065 (original scan 0065.jpg) from resources/seelig/10_circles-and-geometric-mean.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0065 (0065.jpg, Sheet 4 of 4):

Differentiates \(f(t) = -16t^2 + 64t + 80\) to find its maximum \(f(2) = 144\).

\[G = |CH| = \sqrt{m n}, \qquad A_{\text{mean}} = |OC| = \frac{m + n}{2}, \qquad H_{\text{mean}} = |CK| = \frac{2 m n}{m + n}\]

Editorial & Symbolic Notes

Archival scan 0065.jpg from Fred W. Seelig’s notebook dossier on F.10 Circles and Geometric Mean (Page 4 of 4). Companion OpenDocument worksheet(s): General Reference, Harmonic Conjugates. Classical Euclidean foundations: II.14, III.31, VI.8, VI.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter, X(5) — Nine-Point Center, X(6) — Symmedian (Lemoine) Point, X(7) — Gergonne Point, X(8) — Nagel Point.