F.41 Geometric Problems and Extremal Loci — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.41 (Geometric Problems and Extremal Loci) (SeeligNotebooks).

Folio 0257 (Page 1 of 10): Algebraic & Radical Simplification Sheet (001)

[Folio 0257]

Folio 0257 (original scan 0257.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0257 (0257.jpg, Sheet 1 of 10):

Works through radical and quadratic simplifications used across the triangle problem set.

\[A_{\text{acute}} = \int_0^{\sqrt{3}} \left(\sqrt{4 - x^2} - 1\right) dx - \frac{\pi}{4} = \frac{5\pi - 6\sqrt{3}}{12} \approx 0.44297\]

Editorial & Symbolic Notes

Archival scan 0257.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 1 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0258 (Page 2 of 10): Coordinate Line & Circle Intersection Sheet (002)

[Folio 0258]

Folio 0258 (original scan 0258.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0258 (0258.jpg, Sheet 2 of 10):

Solves simultaneous line-circle equations in Cartesian coordinates.

\[p_{\text{accept}} = \frac{A_{\text{acute}}}{\sqrt{3}} \approx 0.25575, \qquad d_{IO} = \sqrt{R^2 - 2 r R}\]

Editorial & Symbolic Notes

Archival scan 0258.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 2 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0259 (Page 3 of 10): Maximizing d_IO for an Acute Triangle — Vesica Integration (2-001, FWS 2015/04/24)

[Folio 0259]

Folio 0259 (original scan 0259.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0259 (0259.jpg, Sheet 3 of 10):

Integrates the acute region \(A = \int_0^{\sqrt{3}}(\sqrt{4-x^2}-1)\,dx - \pi/4 \approx 0.44297\) above the right-triangle semicircle to explain the 26% Monte Carlo hit rate.

\[A_{\text{acute}} = \int_0^{\sqrt{3}} \left(\sqrt{4 - x^2} - 1\right) dx - \frac{\pi}{4} = \frac{5\pi - 6\sqrt{3}}{12} \approx 0.44297\]

Editorial & Symbolic Notes

Archival scan 0259.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 3 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0260 (Page 4 of 10): Extremal Limit C -> (0, 1) for d_IO (2-002)

[Folio 0260]

Folio 0260 (original scan 0260.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0260 (0260.jpg, Sheet 4 of 10):

Tracks \(R, r, d_{IO}\) as \(C\) approaches the boundary point \((0,1)\) of the acute region.

\[p_{\text{accept}} = \frac{A_{\text{acute}}}{\sqrt{3}} \approx 0.25575, \qquad d_{IO} = \sqrt{R^2 - 2 r R}\]

Editorial & Symbolic Notes

Archival scan 0260.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 4 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0261 (Page 5 of 10): Master Geometry Problem Set — Problems 1–4 (2-003)

[Folio 0261]

Folio 0261 (original scan 0261.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0261 (0261.jpg, Sheet 5 of 10):

States research problems on triangle reconstruction from partial elements, Euler’s formula, and harmonic division.

\[A_{\text{acute}} = \int_0^{\sqrt{3}} \left(\sqrt{4 - x^2} - 1\right) dx - \frac{\pi}{4} = \frac{5\pi - 6\sqrt{3}}{12} \approx 0.44297\]

Editorial & Symbolic Notes

Archival scan 0261.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 5 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0262 (Page 6 of 10): Master Geometry Problem Set — Problems 5–9 (2-004)

[Folio 0262]

Folio 0262 (original scan 0262.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0262 (0262.jpg, Sheet 6 of 10):

Poses problems on Apollonian circles, radical axes, and centers of similitude.

\[p_{\text{accept}} = \frac{A_{\text{acute}}}{\sqrt{3}} \approx 0.25575, \qquad d_{IO} = \sqrt{R^2 - 2 r R}\]

Editorial & Symbolic Notes

Archival scan 0262.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 6 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0263 (Page 7 of 10): Master Geometry Problem Set — Problems 10–14 (2-005)

[Folio 0263]

Folio 0263 (original scan 0263.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0263 (0263.jpg, Sheet 7 of 10):

Poses problems on Ceva, Menelaus, Stewart’s Theorem, and barycentric coordinates.

\[A_{\text{acute}} = \int_0^{\sqrt{3}} \left(\sqrt{4 - x^2} - 1\right) dx - \frac{\pi}{4} = \frac{5\pi - 6\sqrt{3}}{12} \approx 0.44297\]

Editorial & Symbolic Notes

Archival scan 0263.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 7 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0264 (Page 8 of 10): Master Geometry Problem Set — Problems 15–19 (2-006)

[Folio 0264]

Folio 0264 (original scan 0264.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0264 (0264.jpg, Sheet 8 of 10):

Poses problems on the nine-point circle, orthic triangle, and Conway’s notation.

\[p_{\text{accept}} = \frac{A_{\text{acute}}}{\sqrt{3}} \approx 0.25575, \qquad d_{IO} = \sqrt{R^2 - 2 r R}\]

Editorial & Symbolic Notes

Archival scan 0264.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 8 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0265 (Page 9 of 10): Master Geometry Problem Set — Problems 20–23 (2-007)

[Folio 0265]

Folio 0265 (original scan 0265.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0265 (0265.jpg, Sheet 9 of 10):

Poses problems on regular polygons, golden triangles, and biquadratic Heron forms.

\[A_{\text{acute}} = \int_0^{\sqrt{3}} \left(\sqrt{4 - x^2} - 1\right) dx - \frac{\pi}{4} = \frac{5\pi - 6\sqrt{3}}{12} \approx 0.44297\]

Editorial & Symbolic Notes

Archival scan 0265.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 9 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0266 (Page 10 of 10): Master Geometry Problem Set — Problems 24–25 (2-008)

[Folio 0266]

Folio 0266 (original scan 0266.jpg) from resources/seelig/41_problems.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0266 (0266.jpg, Sheet 10 of 10):

Concludes the 25-problem syllabus with projective pole-polar and bilinear transformation challenges.

\[p_{\text{accept}} = \frac{A_{\text{acute}}}{\sqrt{3}} \approx 0.25575, \qquad d_{IO} = \sqrt{R^2 - 2 r R}\]

Editorial & Symbolic Notes

Archival scan 0266.jpg from Fred W. Seelig’s notebook dossier on F.41 Geometric Problems and Extremal Loci (Page 10 of 10). Companion OpenDocument worksheet(s): Monte Carlo, General Reference. Classical Euclidean foundations: I.1, III.31. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.