W.08 Law III: The Spread Law & Extended Circumquadrance Law

Exact Rational Proportionality of Vertex Spreads and Opposite Quadrances

In any non-null triangle A_1 A_2 A_3 with side quadrances Q_1, Q_2, Q_3, vertex spreads s_1, s_2, s_3, quadrea A, and circumquadrance K, the ratios of spread to opposite quadrance are equal: s_1 / Q_1 = s_2 / Q_2 = s_3 / Q_3 = A / (4 * Q_1 * Q_2 * Q_3) = 1 / (4 * K).
t c (Q = 4225/64) / A_1 F_1 / (Q = 3136/25) / A_2 F_2 / (Q = 28224/169) A_1 = [0, 0] A_2 = [14, 0] A_3 = [5, 12] F_1 = [224/25, 168/25] F_2 = [350/169, 840/169] C = [7, 33/8] s_i / Q_i = 16/4225 = 1/(4K) K = 4225/64 A1A1 A2A2 A3A3 F1F1 F2F2 CC + − ⟲ Lbl Gde

Interactive GEOMETOR construction for W.08 (Law III: The Spread Law & Extended Circumquadrance Law). Hover any geometric label in the text or figure to highlight the corresponding element.

A_1 : Point A_2 : Point A_3 : Point ===== t = polygon(A_1, A_2, A_3) F_1 = foot(A_1, line(A_2, A_3)) F_2 = foot(A_2, line(A_1, A_3)) C = circumcenter(A_1, A_2, A_3) c = circle(C, A_1)

Mathematical Synthesis

Dropping altitudes to feet F1 and F2 in triangle t gives two right-triangle expressions for the altitude quadrances: \(Q(A_1, F_1) = Q_3 s_2\) and \(Q(A_2, F_2) = Q_3 s_1\), while matching \(Q_2 s_3 = Q_3 s_2\) immediately yields the Spread Law \(s_1 / Q_1 = s_2 / Q_2 = s_3 / Q_3\) (Wildberger2005)—replacing the transcendental Law of Sines with pure field division!

Connecting the Spread Law to the circumcenter C and circumcircle c produces the Extended Spread Law: every ratio \(s_i / Q_i\) equals \(1 / (4 K)\), where \(K = Q(C, A_i) = Q_1 Q_2 Q_3 / \mathcal{A}\) is the circumquadrance—the exact quadratic counterpart of the classical \(a / \sin A = 2 R\) identity.

Key Algebraic Identities

\[\frac{s_1}{Q_1} = \frac{s_2}{Q_2} = \frac{s_3}{Q_3} = \frac{\mathcal{A}}{4 Q_1 Q_2 Q_3} = \frac{1}{4 K}\]
\[K = \frac{Q_1 Q_2 Q_3}{\mathcal{A}}, \qquad \mathcal{A} = 64 K^2 s_1 s_2 s_3\]

Symbolic SymPy Verification

Invariant / Theorem Check

Exact SymPy Value

Verification Status

Quadrances (Q_1, Q_2, Q_3)

(225, 169, 196)

Exact integer quadrances

Vertex spreads (s_1, s_2, s_3)

(144/169, 16/25, 3136/4225)

Exact rational spreads

Common ratio s_1/Q_1 == s_2/Q_2 == s_3/Q_3

16/4225

Verified identical

Circumcenter C & Circumquadrance K

C = [7, 33/8], K = 4225/64

1/(4K) = 16/4225

Cross-Repository Canon

Executable Demonstration

W.08 Law III: The Spread Law & Extended Circumquadrance Law — Code (Standalone model.py & w-08.svg)

Thematic Part

Part II: The Five Fundamental Laws

Euclid’s Elements

III.20, IV.5, VI.4 (EuclidElements)

GEOMETOR Elements

G.3 extremities of a segment, G.5 surface

GEOMETOR Fundamentals

F.11 Circumcenter, F.13 Compare Triangles in Circles

ETC Reference

Circumcenter X(3) (Kimberling1998)

Constructive Dependency Graph

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Depends on: W.04 Spread, Cross, Twist & Line Separation Ratios, W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants, W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea

Required for: W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas, W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle, W.14 Menelaus, Ceva & The Alternate Spreads Theorem, W.15 Circumcenter, Orthocenter & Universal Affine Barycentrics, W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points, W.22 Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge, W.23 Chromogeometry & Universal Symmetric Bilinear Forms, W.24 Exact Triangle Solvers & Surveying Triangulation