W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points

Thales’ Right-Diameter Law, Inscribed Spread s_2(s), and Stereographic Parametrization

On a circle Q(X, O_c) = K over F, a diameter subtends spread 1 at every other circle point (Thales), any chord P_1 P_2 subtends constant spread s_3 at the circumference with Q(P_1, P_2) = 4*K*s_3 and central spread s(O_c P_1, O_c P_2) = S_2(s_3) = 4*s_3*(1 - s_3), and any external or internal point A_0 has invariant secant quadrance product Q(A_0, P_1)*Q(A_0, P_2) = (Q(A_0, O_c) - K)^2.

Interactive GEOMETOR construction for W.17 (Circle Geometry: Subtended Spreads, Secant Products & Rational Points). Hover any geometric label in the text or figure to highlight the corresponding element.

O_c : Point P_1 : Point P_2 : Point B : Point A_0 : Point ===== c = circle(O_c, P_1) l_sec1 = line(A_0, P_1) l_sec2 = line(A_0, B) Q_1 = meet_other(c, l_sec1, P_1) Q_2 = meet_other(c, l_sec2, B)

Mathematical Synthesis

Classical circle theorems suffer from case distinctions between acute and obtuse inscribed angles (\(\theta\) vs. \(\pi - \theta\)). Because supplementary rays define the same line, every point B on circle c (centered at Oc) subtends the exact same spread \(s(B P_1, B P_2) = Q(P_1, P_2) / (4 K)\) on chord \(P_1 P_2\), regardless of which arc B inhabits (Wildberger2005)! And the central spread at Oc is governed by the second spread polynomial \(S_2(s_3) = 4 s_3 (1 - s_3)\).

Similarly, Euclid’s intersecting chords/secants theorem (III.35, III.36) becomes the Equal Products Theorem: lines lsec1 and lsec2 through A0 cut c at (P1, Q1) and (B, Q2) with invariant quadrance product \((Q(A_0, O_c) - K)^2\). Projecting from \([-1, 0]\) parametrizes the unit circle by \(\left[\frac{1 - t^2}{1 + t^2}, \frac{2 t}{1 + t^2}\right]\), generating all Pythagorean triples over \(\mathbb{Q}\) and counting \(p \mp 1\) points on circles over \(\mathbb{F}_p\).

Key Algebraic Identities

\[s(B P_1, B P_2) = \frac{Q(P_1, P_2)}{4 K} = s_3, \qquad s(O_c P_1, O_c P_2) = S_2(s_3) = 4 s_3 (1 - s_3)\]

\[Q(A_0, P_1)\, Q(A_0, Q_1) = Q(A_0, B)\, Q(A_0, Q_2) = \big(Q(A_0, O_c) - K\big)^2, \qquad U(t) = \left[\frac{1 - t^2}{1 + t^2},\; \frac{2 t}{1 + t^2}\right]\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Inscribed spread invariance s(B) == s(Q_1)	1/2 == 1/2	Equals Q(P_1, P_2)/(4K) = 50/100 = 1/2
Central spread s(O_c P_1, O_c P_2)	1	Matches S_2(1/2) = 1
Equal secant products from A_0	275625/256 == 275625/256	Matches (Q(A_0,O_c) - K)^2 = 275625/256
Rational unit circle point count in GF(17), GF(19)	16 (= 17-1), 20 (= 19+1)	p - 1 when -1 square, p + 1 otherwise

Cross-Repository Canon

Executable Demonstration	W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points — Code (Standalone model.py & w-17.svg)
Thematic Part	Part III: Universal Triangle, Conic & Chromogeometry
Euclid’s Elements	III.20, III.21, III.31, III.35, III.36, X.29 (EuclidElements)
GEOMETOR Elements	G.3 extremities of a segment, G.7 plane surface, G.11 center of a circle
GEOMETOR Fundamentals	F.8 Chord Intersection, F.9 Circle Equations, F.33 Euclidean (m, n) Generators
Number Theory Bridge	Counts p + 1 (when -1 non-square) or p - 1 rational points over F_p

Constructive Dependency Graph

Depends on: W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads, W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants, W.08 Law III: The Spread Law & Extended Circumquadrance Law

Required for: W.18 Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation, W.19 Regular Stars, Golden Pentagons & Finite-Field Polygons, W.25 Snellius–Pothenot Resection & Hansen’s Surveying Problem, W.28 Rational Polar Coordinates, Classical Curves & The Ellipson