F.9 Circle Equations — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.9 (Circle Equations) (SeeligNotebooks).

Folio 0054 (Page 1 of 8): Equations of Circles in a Square of Side Z (Part 1)

[Folio 0054]

Folio 0054 (original scan 0054.jpg) from resources/seelig/09_circle-equations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0054 (0054.jpg, Sheet 1 of 8):

Sets up \(x^2+y^2=9\), \(x^2+(y-Z)^2=25\), and \((x-Z)^2+(y-Z)^2=49\) for interior point \(P\) in square \(ABCD\).

\[x^2 + y^2 = 9, \quad x^2 + (y - Z)^2 = 25, \quad (x - Z)^2 + y^2 = 49 \implies 2 Z (Q - P) = 24\]

Editorial & Symbolic Notes

Archival scan 0054.jpg from Fred W. Seelig’s notebook dossier on F.9 Circle Equations (Page 1 of 8). Companion OpenDocument worksheet(s): General Reference, Radical Axes. Classical Euclidean foundations: I.47, III.35.

Folio 0055 (Page 2 of 8): Semicircle Geometric Mean Representation (Part 2)

[Folio 0055]

Folio 0055 (original scan 0055.jpg) from resources/seelig/09_circle-equations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0055 (0055.jpg, Sheet 2 of 8):

Represents \(P^2\) and \(Q^2\) as geometric mean altitudes in semicircles of radii 3, 5, and 7.

\[l_{AE}:\; y = \sqrt{3}x - \frac{\sqrt{2}}{2}R, \qquad P_y = \frac{r_F^2 - r_I^2}{2(F_y - I_y)}\]

Editorial & Symbolic Notes

Archival scan 0055.jpg from Fred W. Seelig’s notebook dossier on F.9 Circle Equations (Page 2 of 8). Companion OpenDocument worksheet(s): General Reference, Radical Axes. Classical Euclidean foundations: I.47, III.35.

Folio 0056 (Page 3 of 8): Pairwise Subtraction of Circle Equations (Part 3)

[Folio 0056]

Folio 0056 (original scan 0056.jpg) from resources/seelig/09_circle-equations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0056 (0056.jpg, Sheet 3 of 8):

Subtracts the circle equations for \(PA=3, PB=5, PD=7\) to obtain \(2Z(Q-P) = 24\).

\[x^2 + y^2 = 9, \quad x^2 + (y - Z)^2 = 25, \quad (x - Z)^2 + y^2 = 49 \implies 2 Z (Q - P) = 24\]

Editorial & Symbolic Notes

Archival scan 0056.jpg from Fred W. Seelig’s notebook dossier on F.9 Circle Equations (Page 3 of 8). Companion OpenDocument worksheet(s): General Reference, Radical Axes. Classical Euclidean foundations: I.47, III.35.

Folio 0057 (Page 4 of 8): Quadratic Solution for P, Q, and Z (FWS 9/24/2014)

[Folio 0057]

Folio 0057 (original scan 0057.jpg) from resources/seelig/09_circle-equations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0057 (0057.jpg, Sheet 4 of 8):

Evaluates \(P = \frac{3}{2}(\sqrt{3}-1) \approx 1.09808, Q = 3\sqrt{\sqrt{3}/2} \approx 2.79181\), and \(Z = 7.08492\).

\[l_{AE}:\; y = \sqrt{3}x - \frac{\sqrt{2}}{2}R, \qquad P_y = \frac{r_F^2 - r_I^2}{2(F_y - I_y)}\]

Editorial & Symbolic Notes

Archival scan 0057.jpg from Fred W. Seelig’s notebook dossier on F.9 Circle Equations (Page 4 of 8). Companion OpenDocument worksheet(s): General Reference, Radical Axes. Classical Euclidean foundations: I.47, III.35.

Folio 0058 (Page 5 of 8): Tony Garcia / Sangaku Problem Setup (Posted Sept. 28, 2014)

[Folio 0058]

Folio 0058 (original scan 0058.jpg) from resources/seelig/09_circle-equations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0058 (0058.jpg, Sheet 5 of 8):

Draws circle \(\Gamma_O(R)\), inscribed square \(ACBD\) of side \(\sqrt{2}R\), equilateral triangle \(\triangle ACE\), and inscribed tangent circle \(\Gamma_P(r)\).

\[x^2 + y^2 = 9, \quad x^2 + (y - Z)^2 = 25, \quad (x - Z)^2 + y^2 = 49 \implies 2 Z (Q - P) = 24\]

Editorial & Symbolic Notes

Archival scan 0058.jpg from Fred W. Seelig’s notebook dossier on F.9 Circle Equations (Page 5 of 8). Companion OpenDocument worksheet(s): General Reference, Radical Axes. Classical Euclidean foundations: I.47, III.35.

Folio 0059 (Page 6 of 8): Line Equation for Side AE of Equilateral Triangle ACE

[Folio 0059]

Folio 0059 (original scan 0059.jpg) from resources/seelig/09_circle-equations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0059 (0059.jpg, Sheet 6 of 8):

Derives \(y = \sqrt{3}x - \frac{\sqrt{2}}{2}R\) for side \(AE\).

\[l_{AE}:\; y = \sqrt{3}x - \frac{\sqrt{2}}{2}R, \qquad P_y = \frac{r_F^2 - r_I^2}{2(F_y - I_y)}\]

Editorial & Symbolic Notes

Archival scan 0059.jpg from Fred W. Seelig’s notebook dossier on F.9 Circle Equations (Page 6 of 8). Companion OpenDocument worksheet(s): General Reference, Radical Axes. Classical Euclidean foundations: I.47, III.35.

Folio 0060 (Page 7 of 8): Circle Equation for Arc CD Centered at B

[Folio 0060]

Folio 0060 (original scan 0060.jpg) from resources/seelig/09_circle-equations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0060 (0060.jpg, Sheet 7 of 8):

Writes \((x - \frac{\sqrt{2}}{2}R)^2 + (y - \frac{\sqrt{2}}{2}R)^2 = 2R^2\) for arc \(\widehat{CD}\).

\[x^2 + y^2 = 9, \quad x^2 + (y - Z)^2 = 25, \quad (x - Z)^2 + y^2 = 49 \implies 2 Z (Q - P) = 24\]

Editorial & Symbolic Notes

Archival scan 0060.jpg from Fred W. Seelig’s notebook dossier on F.9 Circle Equations (Page 7 of 8). Companion OpenDocument worksheet(s): General Reference, Radical Axes. Classical Euclidean foundations: I.47, III.35.

Folio 0061 (Page 8 of 8): Equating r^2 at Tangent Points F and I

[Folio 0061]

Folio 0061 (original scan 0061.jpg) from resources/seelig/09_circle-equations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0061 (0061.jpg, Sheet 8 of 8):

Solves for the Sangaku circle center ordinate \(P_y = \frac{r_F^2 - r_I^2}{2(F_y - I_y)}\).

\[l_{AE}:\; y = \sqrt{3}x - \frac{\sqrt{2}}{2}R, \qquad P_y = \frac{r_F^2 - r_I^2}{2(F_y - I_y)}\]

Editorial & Symbolic Notes

Archival scan 0061.jpg from Fred W. Seelig’s notebook dossier on F.9 Circle Equations (Page 8 of 8). Companion OpenDocument worksheet(s): General Reference, Radical Axes. Classical Euclidean foundations: I.47, III.35.