F.15 Cyclic Quadrilateral

Golden-Ratio Chords for Angles theta, 2*theta, 3*theta, 4*theta, Ptolemy’s Theorem, and Geometric Progression Impossibility

A cyclic quadrilateral WXYZ whose interior angles form the arithmetic progression theta, 2*theta, 3*theta, 4*theta has theta = 36 degrees (W = 36, X = 72, Y = 144, Z = 108) and golden-ratio diagonals |XZ| = 2 sin(36) = sqrt(3 - phi) and |WY| = 2 sin(72) = phi*sqrt(3 - phi) in the unit circle; by contrast, no non-trivial cyclic quadrilateral can have interior angles in geometric progression theta, k*theta, k^2*theta, k^3*theta.

Interactive GEOMETOR construction for F.15 (Cyclic Quadrilateral). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A : Point B : Point C : Point D : Point ===== circumcircle = circle(O, A) ABCD = polygon(A, B, C, D) e = segment(A, C) f = segment(B, D) P = intersect(e, f)

Mathematical Synthesis

Written in April 2015 (SeeligNotebooks), Seelig’s 16-folio investigation of the cyclic quadrilateral \(WXYZ\) explores three distinct problems. First, when the interior angles are \(\theta, 2\theta, 3\theta, 4\theta\), the opposite-angle supplementary law (EuclidElements III.22) forces the ordering \(W = 36^\circ, X = 72^\circ, Y = 144^\circ, Z = 108^\circ\), so the diagonals e and f in a unit circle have exact golden ratio lengths \(|XZ| = 2\sin 36^\circ = \sqrt{3-\phi}\) and \(|YW| = 2\sin 72^\circ = \phi\sqrt{3-\phi}\).

Second, when the central angles are \(\theta, 2\theta, 3\theta, 4\theta\) (\(\theta = 36^\circ\)), the sides and diagonals are \(a = \phi - 1, c = \phi, |WY| = \phi, |XZ| = 2\). Third, in Folios 15–16 (‘An Open :term:`Cyclic Quadrilateral` Problem’), Seelig proves that no non-square cyclic quadrilateral can have interior angles in geometric progression \(\theta, k\theta, k^2\theta, k^3\theta\) because the resulting supplementary cubics \(k^3 \mp k^2 \mp k \pm 1 = 0\) have only roots \(\pm 1, \pm i\).

Key Algebraic Identities

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

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

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.15 Cyclic Quadrilateral — Folios (Folios 0098–0113, 16 scanned manuscript pages in resources/seelig/15_cyclic-quadrilateral)
Euclid’s Elements	III.21, III.22, IV.11 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Cyclic Quad, Chord Intersection (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.8 Chord Intersection