F.15 Cyclic Quadrilateral — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.15 (Cyclic Quadrilateral) (SeeligNotebooks).

Folio 0098 (Page 1 of 16): Cyclic Quad with Angles theta, 2*theta, 3*theta, 4*theta (Cropped Copy, FWS 2015/4/9)

[Folio 0098]

Folio 0098 (original scan 0098.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0098 (0098.jpg, Sheet 1 of 16):

Tabulates the four central isosceles triangles \(\triangle WOX, \triangle XOY, \triangle YOZ, \triangle ZOW\) and sets the goal of finding \(|OP|\).

\[\theta + 2\theta + 3\theta + 4\theta = 360^\circ \implies \theta = 36^\circ, \qquad f = |XZ| = 2\sin 36^\circ = \sqrt{3 - \phi}, \qquad e = |YW| = 2\sin 72^\circ = \phi\sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0098.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 1 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0099 (Page 2 of 16): Cyclic Quad with Angles theta, 2*theta, 3*theta, 4*theta (Full Title, FWS 2015/4/9)

[Folio 0099]

Folio 0099 (original scan 0099.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0099 (0099.jpg, Sheet 2 of 16):

Uncropped master page showing full title and central/inscribed angle relations.

\[a c + b d = e f \quad (\text{Ptolemy}), \qquad (k^3 - k^2 + k - 1) = (k - 1)(k^2 + 1) = 0\]

Editorial & Symbolic Notes

Archival scan 0099.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 2 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0100 (Page 3 of 16): Base Angle Equalities M = K, N = L & Butterfly Similarity (Page 2)

[Folio 0100]

Folio 0100 (original scan 0100.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0100 (0100.jpg, Sheet 3 of 16):

Deduces \(M = K\) and \(N = L\) and applies \(\triangle ZPW \simeq \triangle YPX\).

\[\theta + 2\theta + 3\theta + 4\theta = 360^\circ \implies \theta = 36^\circ, \qquad f = |XZ| = 2\sin 36^\circ = \sqrt{3 - \phi}, \qquad e = |YW| = 2\sin 72^\circ = \phi\sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0100.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 3 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0101 (Page 4 of 16): Inscribed Angles 36, 72, 144, 108 in Unit Circle (FWS Begun April 6, 2015)

[Folio 0101]

Folio 0101 (original scan 0101.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0101 (0101.jpg, Sheet 4 of 16):

Establishes \(\theta = 36^\circ\) and vertex ordering \(W=36^\circ, X=72^\circ, Y=144^\circ, Z=108^\circ\) so opposite angles sum to \(180^\circ\).

\[a c + b d = e f \quad (\text{Ptolemy}), \qquad (k^3 - k^2 + k - 1) = (k - 1)(k^2 + 1) = 0\]

Editorial & Symbolic Notes

Archival scan 0101.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 4 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0102 (Page 5 of 16): Diagonal |XZ| = 2 sin 36 = sqrt(3 - phi) (Page 2)

[Folio 0102]

Folio 0102 (original scan 0102.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0102 (0102.jpg, Sheet 5 of 16):

Computes midpoint distance \(|OM_1| = \cos 36^\circ = \phi/2\) and chord \(|XZ| = \sqrt{3-\phi} \approx 1.1756\).

\[\theta + 2\theta + 3\theta + 4\theta = 360^\circ \implies \theta = 36^\circ, \qquad f = |XZ| = 2\sin 36^\circ = \sqrt{3 - \phi}, \qquad e = |YW| = 2\sin 72^\circ = \phi\sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0102.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 5 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0103 (Page 6 of 16): Power of Diagonal Intersection P (Page 3, citing Vol. I, pp. 265–270)

[Folio 0103]

Folio 0103 (original scan 0103.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0103 (0103.jpg, Sheet 6 of 16):

Relates \(d = |OP|\) to diagonal segments \(m = |PX|, n = |PZ|\) via \(r^2 = d^2 + mn\).

\[a c + b d = e f \quad (\text{Ptolemy}), \qquad (k^3 - k^2 + k - 1) = (k - 1)(k^2 + 1) = 0\]

Editorial & Symbolic Notes

Archival scan 0103.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 6 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0104 (Page 7 of 16): Diagonal |YW| = 2 sin 72 = phi*sqrt(3 - phi) (Page 4)

[Folio 0104]

Folio 0104 (original scan 0104.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0104 (0104.jpg, Sheet 7 of 16):

Computes \(|OM_2| = \cos 72^\circ \approx 0.3090\) and \(|YW| = \phi\sqrt{3-\phi} \approx 1.9021\).

\[\theta + 2\theta + 3\theta + 4\theta = 360^\circ \implies \theta = 36^\circ, \qquad f = |XZ| = 2\sin 36^\circ = \sqrt{3 - \phi}, \qquad e = |YW| = 2\sin 72^\circ = \phi\sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0104.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 7 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0105 (Page 8 of 16): Non-Cyclic Trap When Angles Are Ordered Sequentially

[Folio 0105]

Folio 0105 (original scan 0105.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0105 (0105.jpg, Sheet 8 of 16):

Shows that if \(W=36^\circ, X=72^\circ, Y=108^\circ, Z=144^\circ\) in order around the perimeter, \(W + Y \ne 180^\circ\) so \(WXYZ\) is not cyclic.

\[a c + b d = e f \quad (\text{Ptolemy}), \qquad (k^3 - k^2 + k - 1) = (k - 1)(k^2 + 1) = 0\]

Editorial & Symbolic Notes

Archival scan 0105.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 8 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0106 (Page 9 of 16): Empirical C.a.R. Measurements of Side Lengths & Central Angles

[Folio 0106]

Folio 0106 (original scan 0106.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0106 (0106.jpg, Sheet 9 of 16):

Records C.a.R. values \(a=1.93, b=1.09, c=0.11, d=1.87\) and central angles \(150^\circ, 66^\circ, 6^\circ, 138^\circ\).

\[\theta + 2\theta + 3\theta + 4\theta = 360^\circ \implies \theta = 36^\circ, \qquad f = |XZ| = 2\sin 36^\circ = \sqrt{3 - \phi}, \qquad e = |YW| = 2\sin 72^\circ = \phi\sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0106.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 9 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0107 (Page 10 of 16): Central Angles in Arithmetic Progression (36, 72, 108, 144)

[Folio 0107]

Folio 0107 (original scan 0107.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0107 (0107.jpg, Sheet 10 of 16):

Places vertices at \(e^{j0}, e^{j\theta}, e^{j3\theta}, e^{j6\theta}\), obtaining golden-ratio sides \(a = \phi-1, c = \phi\) and diagonals \(|WY| = \phi, |XZ| = 2\).

\[a c + b d = e f \quad (\text{Ptolemy}), \qquad (k^3 - k^2 + k - 1) = (k - 1)(k^2 + 1) = 0\]

Editorial & Symbolic Notes

Archival scan 0107.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 10 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0108 (Page 11 of 16): Four Central Isosceles Triangles OWX, OXY, OYZ, OZW

[Folio 0108]

Folio 0108 (original scan 0108.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0108 (0108.jpg, Sheet 11 of 16):

Isolated sketches of the four central sector triangles and their base angles.

\[\theta + 2\theta + 3\theta + 4\theta = 360^\circ \implies \theta = 36^\circ, \qquad f = |XZ| = 2\sin 36^\circ = \sqrt{3 - \phi}, \qquad e = |YW| = 2\sin 72^\circ = \phi\sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0108.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 11 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0109 (Page 12 of 16): Vector Construction of the Non-Cyclic Quadrilateral

[Folio 0109]

Folio 0109 (original scan 0109.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0109 (0109.jpg, Sheet 12 of 16):

Constructs the non-cyclic quadrilateral with \(WX = 1\) and comments on ambiguous problem statements.

\[a c + b d = e f \quad (\text{Ptolemy}), \qquad (k^3 - k^2 + k - 1) = (k - 1)(k^2 + 1) = 0\]

Editorial & Symbolic Notes

Archival scan 0109.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 12 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0110 (Page 13 of 16): Ptolemy’s Theorem & Parameterization by Angle mu (Part 1)

[Folio 0110]

Folio 0110 (original scan 0110.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0110 (0110.jpg, Sheet 13 of 16):

Normalizes diagonal \(WY = 1\) and applies the Law of Sines and Ptolemy’s theorem \(ac + bd = |XZ|\).

\[\theta + 2\theta + 3\theta + 4\theta = 360^\circ \implies \theta = 36^\circ, \qquad f = |XZ| = 2\sin 36^\circ = \sqrt{3 - \phi}, \qquad e = |YW| = 2\sin 72^\circ = \phi\sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0110.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 13 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0111 (Page 14 of 16): Product-to-Sum Reduction for |XZ| (Part 2)

[Folio 0111]

Folio 0111 (original scan 0111.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0111 (0111.jpg, Sheet 14 of 16):

Expresses \(|XZ|\) in terms of \(\cos(2\mu - 72^\circ)\) and \(\cos(252^\circ - 2\mu)\).

\[a c + b d = e f \quad (\text{Ptolemy}), \qquad (k^3 - k^2 + k - 1) = (k - 1)(k^2 + 1) = 0\]

Editorial & Symbolic Notes

Archival scan 0111.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 14 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0112 (Page 15 of 16): An Open Cyclic Quadrilateral Problem (Fred Seelig, April 2015)

[Folio 0112]

Folio 0112 (original scan 0112.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0112 (0112.jpg, Sheet 15 of 16):

Poses the question of whether a cyclic quadrilateral can have interior angles in geometric progression \(\theta, k\theta, k^2\theta, k^3\theta\).

\[\theta + 2\theta + 3\theta + 4\theta = 360^\circ \implies \theta = 36^\circ, \qquad f = |XZ| = 2\sin 36^\circ = \sqrt{3 - \phi}, \qquad e = |YW| = 2\sin 72^\circ = \phi\sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0112.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 15 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.

Folio 0113 (Page 16 of 16): Impossibility Proof via Cubic Factorization

[Folio 0113]

Folio 0113 (original scan 0113.jpg) from resources/seelig/15_cyclic-quadrilateral.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0113 (0113.jpg, Sheet 16 of 16):

Proves that \(k^3 - k^2 - k + 1 = 0\), \(k^3 - k^2 + k - 1 = 0\), and \(k^3 + k^2 - k - 1 = 0\) have only roots \(\pm 1, \pm i\), ruling out any non-trivial geometric progression.

\[a c + b d = e f \quad (\text{Ptolemy}), \qquad (k^3 - k^2 + k - 1) = (k - 1)(k^2 + 1) = 0\]

Editorial & Symbolic Notes

Archival scan 0113.jpg from Fred W. Seelig’s notebook dossier on F.15 Cyclic Quadrilateral (Page 16 of 16). Companion OpenDocument worksheet(s): Cyclic Quad, Chord Intersection. Classical Euclidean foundations: III.21, III.22, IV.11.