F.11 Circumcenter — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.11 (Circumcenter) (SeeligNotebooks).

Folio 0066 (Page 1 of 6): Circumcenter of a Triangle (FWS 5/31/2011)

[Folio 0066]

Folio 0066 (original scan 0066.jpg) from resources/seelig/11_circumcenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0066 (0066.jpg, Sheet 1 of 6):

Summary sheet with circumcircle diagram and Cartesian formulas for \(a, b, c, s, A_T, R = abc/(4A_T)\), and \((x_0, y_0)\).

\[R = \frac{a b c}{4 K}, \qquad O = \left(a^2 S_A : b^2 S_B : c^2 S_C\right) = \left(\sin 2\alpha : \sin 2\beta : \sin 2\gamma\right)\]

Editorial & Symbolic Notes

Archival scan 0066.jpg from Fred W. Seelig’s notebook dossier on F.11 Circumcenter (Page 1 of 6). Companion OpenDocument worksheet(s): Circumcircle-Circumcenter, Circumcircle Solver, General Reference. Classical Euclidean foundations: III.1, IV.5. Kimberling Encyclopedia of Triangle Centers loci: X(3) — Circumcenter, X(26) — Circumcenter of the Tangential Triangle.

Folio 0067 (Page 2 of 6): Poem by Christopher Morley (1890–1957)

[Folio 0067]

Folio 0067 (original scan 0067.jpg) from resources/seelig/11_circumcenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0067 (0067.jpg, Sheet 2 of 6):

Handwritten transcription of Morley’s 6-line geometric poem ‘A circle is a happy thing to be’.

\[D = 2\left[x_A(y_C - y_B) + x_B(y_A - y_C) + x_C(y_B - y_A)\right], \qquad x_0 = \frac{\sum (x_A^2 + y_A^2)(y_C - y_B)}{D}\]

Editorial & Symbolic Notes

Archival scan 0067.jpg from Fred W. Seelig’s notebook dossier on F.11 Circumcenter (Page 2 of 6). Companion OpenDocument worksheet(s): Circumcircle-Circumcenter, Circumcircle Solver, General Reference. Classical Euclidean foundations: III.1, IV.5. Kimberling Encyclopedia of Triangle Centers loci: X(3) — Circumcenter, X(26) — Circumcenter of the Tangential Triangle.

Folio 0068 (Page 3 of 6): Sign Correction of O’Rourke’s Formula (FWS 5/25/2011)

[Folio 0068]

Folio 0068 (original scan 0068.jpg) from resources/seelig/11_circumcenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0068 (0068.jpg, Sheet 3 of 6):

Derives \((O_x, O_y)\) from perpendicular bisector intersections and corrects the sign error in Joseph O’Rourke’s formula.

\[R = \frac{a b c}{4 K}, \qquad O = \left(a^2 S_A : b^2 S_B : c^2 S_C\right) = \left(\sin 2\alpha : \sin 2\beta : \sin 2\gamma\right)\]

Editorial & Symbolic Notes

Archival scan 0068.jpg from Fred W. Seelig’s notebook dossier on F.11 Circumcenter (Page 3 of 6). Companion OpenDocument worksheet(s): Circumcircle-Circumcenter, Circumcircle Solver, General Reference. Classical Euclidean foundations: III.1, IV.5. Kimberling Encyclopedia of Triangle Centers loci: X(3) — Circumcenter, X(26) — Circumcenter of the Tangential Triangle.

Folio 0069 (Page 4 of 6): Circumcircle Subtriangles

[Folio 0069]

Folio 0069 (original scan 0069.jpg) from resources/seelig/11_circumcenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0069 (0069.jpg, Sheet 4 of 6):

Decomposes \(K = K_A + K_B + K_C = \frac{1}{2}\sum a\sqrt{R^2 - (a/2)^2}\) using central angles \(\Phi_C = 2\gamma\) and subtriangle semiperimeters \(s_C = (2R+c)/2\).

\[D = 2\left[x_A(y_C - y_B) + x_B(y_A - y_C) + x_C(y_B - y_A)\right], \qquad x_0 = \frac{\sum (x_A^2 + y_A^2)(y_C - y_B)}{D}\]

Editorial & Symbolic Notes

Archival scan 0069.jpg from Fred W. Seelig’s notebook dossier on F.11 Circumcenter (Page 4 of 6). Companion OpenDocument worksheet(s): Circumcircle-Circumcenter, Circumcircle Solver, General Reference. Classical Euclidean foundations: III.1, IV.5. Kimberling Encyclopedia of Triangle Centers loci: X(3) — Circumcenter, X(26) — Circumcenter of the Tangential Triangle.

Folio 0070 (Page 5 of 6): Tangential Triangle Drawings (Acute vs. Obtuse)

[Folio 0070]

Folio 0070 (original scan 0070.jpg) from resources/seelig/11_circumcenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0070 (0070.jpg, Sheet 5 of 6):

Side-by-side compass-and-straightedge constructions of the tangential triangle for an acute triangle and an obtuse triangle.

\[R = \frac{a b c}{4 K}, \qquad O = \left(a^2 S_A : b^2 S_B : c^2 S_C\right) = \left(\sin 2\alpha : \sin 2\beta : \sin 2\gamma\right)\]

Editorial & Symbolic Notes

Archival scan 0070.jpg from Fred W. Seelig’s notebook dossier on F.11 Circumcenter (Page 5 of 6). Companion OpenDocument worksheet(s): Circumcircle-Circumcenter, Circumcircle Solver, General Reference. Classical Euclidean foundations: III.1, IV.5. Kimberling Encyclopedia of Triangle Centers loci: X(3) — Circumcenter, X(26) — Circumcenter of the Tangential Triangle.

Folio 0071 (Page 6 of 6): Properties of the Tangential Triangle

[Folio 0071]

Folio 0071 (original scan 0071.jpg) from resources/seelig/11_circumcenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0071 (0071.jpg, Sheet 6 of 6):

Explanatory notes defining the tangential triangle formed by the tangents to the circumcircle at \(A, B, C\).

\[D = 2\left[x_A(y_C - y_B) + x_B(y_A - y_C) + x_C(y_B - y_A)\right], \qquad x_0 = \frac{\sum (x_A^2 + y_A^2)(y_C - y_B)}{D}\]

Editorial & Symbolic Notes

Archival scan 0071.jpg from Fred W. Seelig’s notebook dossier on F.11 Circumcenter (Page 6 of 6). Companion OpenDocument worksheet(s): Circumcircle-Circumcenter, Circumcircle Solver, General Reference. Classical Euclidean foundations: III.1, IV.5. Kimberling Encyclopedia of Triangle Centers loci: X(3) — Circumcenter, X(26) — Circumcenter of the Tangential Triangle.