W.23 Chromogeometry & Universal Symmetric Bilinear Forms

Harmonizing Blue Euclidean, Red Lorentzian, and Green Hyperbolic Triangle Centers

Replacing the Euclidean quadratic form with any non-degenerate symmetric bilinear form M on F^2 preserves all Five Laws of Rational Trigonometry; in Chromogeometry—pairing Blue (dx^2 + dy^2), Red (dx^2 - dy^2), and Green (2*dx*dy) metrics on the same triangle—the three Euler lines concur at the centroid G, C_B C_R O_B C_G is a parallelogram, and N_B = (C_R + C_G)/2.
t p_chromo e_B e_R e_G A_1 = [0, 0] A_2 = [8, 2] A_3 = [2, 6] G = [10/3, 8/3] C_B = [41/11, 23/11] O_B = [28/11, 42/11] N_B = [69/22, 65/22] C_R = [53/11, 47/11] O_R = [4/11, -6/11] C_G = [16/11, 18/11] O_G = [78/11, 52/11] Q_B^2 = Q_R^2 + Q_G^2: 68^2 = 60^2 + 32^2 N_B = (C_R + C_G)/2, Euler lines concur at G A1A1 A2A2 A3A3 GG CBCB OBOB NBNB CRCR OROR CGCG OGOG + − ⟲ Lbl Gde

Interactive GEOMETOR construction for W.23 (Chromogeometry & Universal Symmetric Bilinear Forms). 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) G = centroid(A_1, A_2, A_3) C_B, O_B, N_B = centers_blue(A_1, A_2, A_3) C_R, O_R = centers_red(A_1, A_2, A_3) C_G, O_G = centers_green(A_1, A_2, A_3) e_B = line(C_B, O_B) e_R = line(C_R, O_R) e_G = line(C_G, O_G) p_chromo = polygon(C_B, C_R, O_B, C_G)

Mathematical Synthesis

Because Rational Trigonometry uses only polynomial identities, every theorem in Parts I–III generalizes to any symmetric bilinear form \(u \cdot_M v = u^T M v\) (\(\det M \neq 0\)) with \(s_M(\ell_1, \ell_2)\) scaled by \(\det M\) (Wildberger2005, Wildberger2008)! In Norman J. Wildberger’s Chromogeometry, three canonical quadratic forms act simultaneously on the same planar triangle t: Blue Euclidean \(Q_B = \Delta x^2 + \Delta y^2\), Red Lorentz/Minkowski \(Q_R = \Delta x^2 - \Delta y^2\), and Green rotated Lorentz \(Q_G = 2 \Delta x \Delta y\), satisfying Pythagoras’ Identity in Metrical Space \(Q_B^2 = Q_R^2 + Q_G^2\)!

Computing the circumcenters (CB, CR, CG), orthocenters (OB, OR, OG), and Euler lines (eB, eR, eG) uncovers miraculous cross-metric harmonies: all three colored Euler lines concur at the metric-free centroid G; the quadrilateral pchromo (\(C_B C_R O_B C_G\)) is always a parallelogram (\(C_B + O_B = C_R + C_G\)); and the Blue nine-point center NB is the exact midpoint of the Red and Green circumcenters!

Key Algebraic Identities

\[Q_B = \Delta x^2 + \Delta y^2, \quad Q_R = \Delta x^2 - \Delta y^2, \quad Q_G = 2 \Delta x \Delta y \implies Q_B^2 = Q_R^2 + Q_G^2\]
\[C_B + O_B = C_R + C_G, \qquad N_B = \frac{C_R + C_G}{2}, \qquad N_R = \frac{C_B + C_G}{2}, \qquad N_G = \frac{C_B + C_R}{2}\]

Symbolic SymPy Verification

Invariant / Theorem Check

Exact SymPy Value

Verification Status

Pythagoras in Metrical Space Q_B^2 == Q_R^2 + Q_G^2

68^2 == 60^2 + 32^2

68^2 = 60^2 + 32^2 = 4624

Nine-Point Midpoint N_B == (C_R + C_G)/2

[69/22, 65/22] == [69/22, 65/22]

Chromo-Nine-Point Theorem verified

Parallelogram C_B + O_B == C_R + C_G

2 * N_B = [69/11, 65/11]

C_B C_R O_B C_G is a parallelogram

Concurrency of Blue, Red, Green Euler lines

G = [10/3, 8/3]

All 3 Euler lines pass through X(2)

Cross-Repository Canon

Executable Demonstration

W.23 Chromogeometry & Universal Symmetric Bilinear Forms — Code (Standalone model.py & w-23.svg)

Thematic Part

Part III: Universal Triangle, Conic & Chromogeometry

Relativistic Canon

Hermann Minkowski (1908 spacetime metric dx^2 - dt^2)

Chromogeometry

Wildberger2008 (Chromogeometry, 12+ colored triangle center theorems)

GEOMETOR Fundamentals

F.6 Centroid, F.34 Nine-Point Circle

Universal Invariance

All Five Laws hold verbatim for Blue, Red, Green, and arbitrary M = M^T

Constructive Dependency Graph

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Depends on: W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants, W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea, W.08 Law III: The Spread Law & Extended Circumquadrance Law, W.09 Law IV: The Cross Law & Quadrea-Spread Identity, W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas, W.16 The Euler Line & Nine-Point Circle Over Arbitrary Fields

Required for: W.26 Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity