F.43 Radical Axes and Radical Center — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.43 (Radical Axes and Radical Center) (SeeligNotebooks).

Folio 0271 (Page 1 of 11): Two-Circle Radical Axis & Equal Tangents (001)

[Folio 0271]

Folio 0271 (original scan 0271.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0271 (0271.jpg, Sheet 1 of 11):

Derives \(x = \frac{d}{2} + \frac{r_1^2-r_2^2}{2d}\) and tests \(r_1=1, r_2=2, d=5 \implies x=2.2, x'=2.8, t = \frac{4\sqrt{6}}{5}\).

\[x = |O_1 P| = \frac{d}{2} + \frac{r_1^2 - r_2^2}{2 d}, \qquad x' = |O_2 P| = \frac{d}{2} - \frac{r_1^2 - r_2^2}{2 d}, \qquad t^2 = x^2 - r_1^2 = {x'}^2 - r_2^2\]

Editorial & Symbolic Notes

Archival scan 0271.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 1 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0272 (Page 2 of 11): Radical Center of Vertex-Altitude Circles — Setup (002, FWS Dec. 8/2014)

[Folio 0272]

Folio 0272 (original scan 0272.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0272 (0272.jpg, Sheet 2 of 11):

Defines circles \(\Gamma_A(h_A), \Gamma_B(h_B), \Gamma_C(h_C)\) centered at \(A, B, C\) with radii equal to the three altitudes.

\[\Gamma_A(A, h_A),\; \Gamma_B(B, h_B),\; \Gamma_C(C, h_C) \implies \frac{x}{x_0} + \frac{y}{y_0} = 1\]

Editorial & Symbolic Notes

Archival scan 0272.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 2 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0273 (Page 3 of 11): Radical Axis of Gamma_A and Gamma_B (003)

[Folio 0273]

Folio 0273 (original scan 0273.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0273 (0273.jpg, Sheet 3 of 11):

Computes the radical foot along side \(c = AB\) and writes the perpendicular line equation.

\[x = |O_1 P| = \frac{d}{2} + \frac{r_1^2 - r_2^2}{2 d}, \qquad x' = |O_2 P| = \frac{d}{2} - \frac{r_1^2 - r_2^2}{2 d}, \qquad t^2 = x^2 - r_1^2 = {x'}^2 - r_2^2\]

Editorial & Symbolic Notes

Archival scan 0273.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 3 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0274 (Page 4 of 11): Radical Axis of Gamma_A and Gamma_C in Intercept Form (004)

[Folio 0274]

Folio 0274 (original scan 0274.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0274 (0274.jpg, Sheet 4 of 11):

Projects the radical axis of \(\Gamma_A\) and \(\Gamma_C\) onto the Cartesian frame of \(AB\) and derives \(x/x_0 + y/y_0 = 1\).

\[\Gamma_A(A, h_A),\; \Gamma_B(B, h_B),\; \Gamma_C(C, h_C) \implies \frac{x}{x_0} + \frac{y}{y_0} = 1\]

Editorial & Symbolic Notes

Archival scan 0274.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 4 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0275 (Page 5 of 11): Radical Axis of Gamma_B and Gamma_C (005)

[Folio 0275]

Folio 0275 (original scan 0275.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0275 (0275.jpg, Sheet 5 of 11):

Derives the double-intercept equation for the third radical axis opposite side \(a = BC\).

\[x = |O_1 P| = \frac{d}{2} + \frac{r_1^2 - r_2^2}{2 d}, \qquad x' = |O_2 P| = \frac{d}{2} - \frac{r_1^2 - r_2^2}{2 d}, \qquad t^2 = x^2 - r_1^2 = {x'}^2 - r_2^2\]

Editorial & Symbolic Notes

Archival scan 0275.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 5 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0276 (Page 6 of 11): 2x2 Linear System for the Radical Center W (006)

[Folio 0276]

Folio 0276 (original scan 0276.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0276 (0276.jpg, Sheet 6 of 11):

Intersects two radical axes in intercept form to solve for the radical center \(W = (W_x, W_y)\).

\[\Gamma_A(A, h_A),\; \Gamma_B(B, h_B),\; \Gamma_C(C, h_C) \implies \frac{x}{x_0} + \frac{y}{y_0} = 1\]

Editorial & Symbolic Notes

Archival scan 0276.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 6 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0277 (Page 7 of 11): Numerical Evaluation on the (2, 3, 4) Triangle — Part 1 (007)

[Folio 0277]

Folio 0277 (original scan 0277.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0277 (0277.jpg, Sheet 7 of 11):

Computes the altitudes \(h_A, h_B, h_C\) and pairwise radical intercepts of the \((2,3,4)\) obtuse triangle.

\[x = |O_1 P| = \frac{d}{2} + \frac{r_1^2 - r_2^2}{2 d}, \qquad x' = |O_2 P| = \frac{d}{2} - \frac{r_1^2 - r_2^2}{2 d}, \qquad t^2 = x^2 - r_1^2 = {x'}^2 - r_2^2\]

Editorial & Symbolic Notes

Archival scan 0277.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 7 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0278 (Page 8 of 11): Numerical Evaluation on the (2, 3, 4) Triangle — Part 2 (008)

[Folio 0278]

Folio 0278 (original scan 0278.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0278 (0278.jpg, Sheet 8 of 11):

Finds the exact radical center coordinates \(W = (2.586, 0.603)\) and verifies concurrency of all three radical axes.

\[\Gamma_A(A, h_A),\; \Gamma_B(B, h_B),\; \Gamma_C(C, h_C) \implies \frac{x}{x_0} + \frac{y}{y_0} = 1\]

Editorial & Symbolic Notes

Archival scan 0278.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 8 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0279 (Page 9 of 11): Two-Circle Separation Regimes (009)

[Folio 0279]

Folio 0279 (original scan 0279.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0279 (0279.jpg, Sheet 9 of 11):

Classifies disjoint, externally tangent, intersecting, internally tangent, and nested circle pairs by comparing \(d\) with \(r_1 \pm r_2\).

\[x = |O_1 P| = \frac{d}{2} + \frac{r_1^2 - r_2^2}{2 d}, \qquad x' = |O_2 P| = \frac{d}{2} - \frac{r_1^2 - r_2^2}{2 d}, \qquad t^2 = x^2 - r_1^2 = {x'}^2 - r_2^2\]

Editorial & Symbolic Notes

Archival scan 0279.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 9 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0280 (Page 10 of 11): Common Chord & Sagitta of Intersecting Circles (010)

[Folio 0280]

Folio 0280 (original scan 0280.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0280 (0280.jpg, Sheet 10 of 11):

Expresses the common chord length \(2y = 2\sqrt{r_1^2 - x^2}\) in terms of Heron’s radical on \((r_1, r_2, d)\).

\[\Gamma_A(A, h_A),\; \Gamma_B(B, h_B),\; \Gamma_C(C, h_C) \implies \frac{x}{x_0} + \frac{y}{y_0} = 1\]

Editorial & Symbolic Notes

Archival scan 0280.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 10 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.

Folio 0281 (Page 11 of 11): Circular Lens Area Formula (011)

[Folio 0281]

Folio 0281 (original scan 0281.jpg) from resources/seelig/43_radical-axes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0281 (0281.jpg, Sheet 11 of 11):

Integrates/sums the two circular sector-minus-triangle segments to obtain the asymmetric lens area.

\[x = |O_1 P| = \frac{d}{2} + \frac{r_1^2 - r_2^2}{2 d}, \qquad x' = |O_2 P| = \frac{d}{2} - \frac{r_1^2 - r_2^2}{2 d}, \qquad t^2 = x^2 - r_1^2 = {x'}^2 - r_2^2\]

Editorial & Symbolic Notes

Archival scan 0281.jpg from Fred W. Seelig’s notebook dossier on F.43 Radical Axes and Radical Center (Page 11 of 11). Companion OpenDocument worksheet(s): Radical Axes, Coaxial Circles. Classical Euclidean foundations: II.5, III.36.