W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea

Polynomial Collinearity Criterion and Radical-Free Triangle Area Algebra

Three planar points A_1, A_2, A_3 with pairwise quadrances Q_1, Q_2, Q_3 are collinear if and only if Archimedes’ symmetric quadratic form A(Q_1, Q_2, Q_3) = (Q_1 + Q_2 + Q_3)^2 - 2(Q_1^2 + Q_2^2 + Q_3^2) vanishes; for any triangle, A(Q_1, Q_2, Q_3) equals 4 times the squared coordinate determinant (16 * Delta^2), defining the triangle’s quadrea.

Interactive GEOMETOR construction for W.07 (Law II: Triple Quad Formula, Archimedes’ Function & Quadrea). Hover any geometric label in the text or figure to highlight the corresponding element.

A_1 : Point A_2 : Point A_3 : Point P_col : Point ===== t = polygon(A_1, A_2, A_3) l_base = line(A_1, A_2) h_3 = segment(A_3, foot(A_3, l_base))

Mathematical Synthesis

In classical geometry, collinearity is expressed linearly via distances \(d_1 + d_2 = d_3\) (which depends on ordering), while triangle area uses Heron’s radical \(\Delta = \sqrt{s(s-a)(s-b)(s-c)}\). Squaring out all three sign combinations \((\pm\sqrt{Q_1} \pm \sqrt{Q_2} \pm \sqrt{Q_3}) = 0\) yields Archimedes’ symmetric polynomial \(\mathcal{A}(Q_1, Q_2, Q_3)\) (Wildberger2005).

Over any field \(F\) (\(\operatorname{char} F \neq 2\)), the Triple Quad Formula \(\mathcal{A}(Q_1, Q_2, Q_3) = 0\) characterizes collinearity (illustrated by Pcol on lbase) without ordering or square roots! When A1, A2, A3 form a triangle t with altitude h3, the quadrea \(\mathcal{A} = 4 Q_3 Q(A_3, \ell_{\text{base}}) = 16 \Delta^2\) is always an element of \(F\)—for the Heronian \((13, 14, 15)\) triangle with \(Q = (169, 196, 225)\), \(\mathcal{A} = 112896 = 16 \times 84^2\).

Key Algebraic Identities

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

\[\begin{split}A_1, A_2, A_3 \text{ collinear} \iff \mathcal{A}(Q_1, Q_2, Q_3) = 0, \qquad \mathcal{A}(Q_1, Q_2, Q_3) = 4 \det\begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix}^2 = 16\Delta^2\end{split}\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Collinear quadrances (A_1, P_col, A_2)	(81, 25, 196)	A(81, 25, 196) = 0
Heronian triangle quadrances (Q_1, Q_2, Q_3)	(225, 169, 196)	Sides (15, 13, 14) with zero radicals
Triangle quadrea A(Q_1, Q_2, Q_3)	112896	Matches 4*det^2 = 112896
Altitude relation 4 * Q_3 * Q(A_3, l_base)	112896	112896 == A

Cross-Repository Canon

Executable Demonstration	W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea — Code (Standalone model.py & w-07.svg)
Thematic Part	Part II: The Five Fundamental Laws
Euclid’s Elements	II.12, II.13 (EuclidElements)
GEOMETOR Elements	G.1 point, G.3 extremities of a segment
GEOMETOR Fundamentals	F.3 Biquadratic Forms in Heron’s Formula (SeeligNotebooks)
ETC Reference	Acute Heronian reference triangle (13, 14, 15)

Constructive Dependency Graph

Depends on: W.01 Planar Points, Line Proportions & Incidence Determinants, W.03 Quadrance, Midpoints & Point-to-Line Separation, W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants

Required for: W.08 Law III: The Spread Law & Extended Circumquadrance Law, W.09 Law IV: The Cross Law & Quadrea-Spread Identity, W.13 Stewart’s Theorem, Median Quadrances & The Centroid, W.18 Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation, W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas, W.23 Chromogeometry & Universal Symmetric Bilinear Forms, W.27 3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads