W.18 Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation

Opposite-Spread Equality, Radical-Free Brahmagupta/Robbins, and Determinant Cocircularity

Four non-collinear points A_1, A_2, A_3, A_4 on a circle c satisfy equal opposite spreads s(A_1 A_2, A_1 A_4) = s(A_3 A_2, A_3 A_4), Ptolemy’s Triple-Quad Law A(Q_12*Q_34, Q_23*Q_14, Q_13*Q_24) = 0 among product quadrances, and the cyclic quadrea / circumquadrance formulas, while four collinear points satisfy Q(Q_12, Q_23, Q_34, Q_14) = 0.

Interactive GEOMETOR construction for W.18 (Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation). Hover any geometric label in the text or figure to highlight the corresponding element.

O_c : Point A_1 : Point A_2 : Point A_3 : Point A_4 : Point ===== c = circle(O_c, A_1) q_cyc = polygon(A_1, A_2, A_3, A_4) d_13 = segment(A_1, A_3) d_24 = segment(A_2, A_4) P_x = meet(line(A_1, A_3), line(A_2, A_4))

Mathematical Synthesis

In Euclid’s III.22, opposite angles of a cyclic quadrilateral sum to two right angles (\(\theta_1 + \theta_3 = \pi\)). Taking \(\sin^2\) turns supplementarity into exact equality: on circle c (center Oc), cyclic quadrilateral qcyc with diagonals d13 and d24 meeting at Px has equal opposite vertex spreads \(s_1 = s_3\) and \(s_2 = s_4\) (Wildberger2005)!

Equally remarkable is Claudius Ptolemy’s theorem (\(ac + bd = ef\) in lengths): squaring out the radicals shows that the three product pairs \((Q_{12} Q_{34},\; Q_{23} Q_{14},\; Q_{13} Q_{24})\) obey the exact Triple Quad Formula \(\mathcal{A} = 0\)! Meanwhile, Leonhard Euler’s four-point relation proves that for any four coplanar points, the Cayley-Menger determinant vanishes identically, and for four collinear points their sequential quadrances satisfy the Quadruple Quad Formula \(Q(Q_{12}, Q_{23}, Q_{34}, Q_{14}) = 0\).

Key Algebraic Identities

\[\text{Cyclic } A_1 A_2 A_3 A_4:\quad s_1 = s_3, \quad s_2 = s_4, \qquad \mathcal{A}(Q_{12} Q_{34},\; Q_{23} Q_{14},\; Q_{13} Q_{24}) = 0\]

\[Q(a, b, c, d) = \big((a + b + c + d)^2 - 2(a^2 + b^2 + c^2 + d^2)\big)^2 - 64 a b c d = 0\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Opposite spreads s_1 == s_3 & s_2 == s_4	1 == 1, 9/10 == 9/10	Theorem 108 verified
Ptolemy's Theorem A(Q_12*Q_34, Q_23*Q_14, Q_13*Q_24)	0	Verified == 0
Euler's 4-Point Cayley-Menger E(Q_ij)	0	Verified == 0
Collinear Quadruple Quad Q(4, 9, 25, 100)	0	Verified == 0

Cross-Repository Canon

Executable Demonstration	W.18 Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation — Code (Standalone model.py & w-18.svg)
Thematic Part	Part III: Universal Triangle, Conic & Chromogeometry
Euclid’s Elements	III.21, III.22 (EuclidElements)
GEOMETOR Elements	G.3 extremities of a segment, G.7 plane surface
GEOMETOR Fundamentals	F.15 Cyclic Quadrilateral (16 manuscript folios, SeeligNotebooks)
Classical Canon	Brahmagupta & Claudius Ptolemy (Almagest)

Constructive Dependency Graph

Depends on: W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea, W.09 Law IV: The Cross Law & Quadrea-Spread Identity, W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points