W.09 Law IV: The Cross Law & Quadrea-Spread Identity

Quadratic Replacement for the Law of Cosines Without Ambiguous Angle Signs

In any triangle A_1 A_2 A_3 with non-null sides Q_1, Q_2 meeting with spread s_3 and cross c_3 = 1 - s_3 at vertex A_3, the quadrances satisfy (Q_1 + Q_2 - Q_3)^2 = 4 * Q_1 * Q_2 * (1 - s_3) = 4 * Q_1 * Q_2 * c_3, equivalently A(Q_1, Q_2, Q_3) = 4 * Q_1 * Q_2 * s_3.
t l_1 alt_1 (Q = 784/41) A_1 = [0, 0] A_2 = [7, 0] A_3 = [2, 4] F_1 = [112/41, 140/41] (Q_1+Q_2-Q_3)^2 = 144 = 4*Q_1*Q_2*c_3 s_3 = 196/205, c_3 = 9/205 A1A1 A2A2 A3A3 F1F1 + − ⟲ Lbl Gde

Interactive GEOMETOR construction for W.09 (Law IV: The Cross Law & Quadrea-Spread Identity). 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) l_1 = line(A_2, A_3) F_1 = foot(A_1, l_1) alt_1 = segment(A_1, F_1)

Mathematical Synthesis

Squaring the Conway-style dot-product term \(Q_1 + Q_2 - Q_3\) eliminates the sign ambiguity between acute and obtuse angles in the classical Law of Cosines, producing the Cross Law \((Q_1 + Q_2 - Q_3)^2 = 4 Q_1 Q_2 (1 - s_3)\) (Wildberger2005). Geometrically, dropping altitude alt1 from A1 to foot F1 on line l1 decomposes triangle t into two right triangles whose Pythagorean relations eliminate \(Q(A_3, F_1)\) quadratically.

Subtracting \(4 Q_1 Q_2\) from both sides reveals the Quadrea Spread Theorem \(\mathcal{A} = 4 Q_1 Q_2 s_3\) (the rational analogue of \(\Delta = \frac{1}{2} a b \sin C\)). Given rational quadrances—such as \(Q = (16, 49, 25)\) for sides \((4, 7, 5)\)—the Cross Law immediately yields exact rational spreads \(s_1 = 384/1225\), \(s_2 = 24/25\), and \(s_3 = 24/49\) by elementary arithmetic alone!

Key Algebraic Identities

\[(Q_1 + Q_2 - Q_3)^2 = 4 Q_1 Q_2 (1 - s_3) = 4 Q_1 Q_2 c_3\]
\[s_3 = 1 - \frac{(Q_1 + Q_2 - Q_3)^2}{4 Q_1 Q_2} = \frac{\mathcal{A}(Q_1, Q_2, Q_3)}{4 Q_1 Q_2}\]

Symbolic SymPy Verification

Invariant / Theorem Check

Exact SymPy Value

Verification Status

Side quadrances (Q_1, Q_2, Q_3)

(41, 20, 49)

Irrational lengths, integer quadrances!

Cross Law LHS (Q_1 + Q_2 - Q_3)^2

144

Matches RHS 4*Q_1*Q_2*(1-s_3) = 144

Recovered rational spreads (s_1, s_2, s_3)

(4/5, 16/41, 196/205)

Pure rational division

Quadrea-Spread identity 4*Q_1*Q_2*s_3

3136

Matches quadrea A = 3136

Cross-Repository Canon

Executable Demonstration

W.09 Law IV: The Cross Law & Quadrea-Spread Identity — Code (Standalone model.py & w-09.svg)

Thematic Part

Part II: The Five Fundamental Laws

Euclid’s Elements

II.12, II.13 (Obtuse and acute generalizations of Pythagoras)

GEOMETOR Elements

G.3 extremities of a segment, G.4 segment

GEOMETOR Fundamentals

F.14 Conway Notation (\(4 S_C^2 = (a^2 + b^2 - c^2)^2\))

Classical Bridge

Replaces c^2 = a^2 + b^2 - 2ab cos C over arbitrary fields

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.13 Stewart’s Theorem, Median Quadrances & The Centroid, W.15 Circumcenter, Orthocenter & Universal Affine Barycentrics, W.18 Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation, W.23 Chromogeometry & Universal Symmetric Bilinear Forms, W.24 Exact Triangle Solvers & Surveying Triangulation