W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle

Pons Asinorum, the 3/4 Equilateral Spread, and Pure Golden Powers Q = (1, phi, phi^2)

In a non-null triangle, Q_1 = Q_2 iff s_1 = s_2 (with apex spread s_3 = (Q_3/Q_1)(1 - Q_3/(4*Q_1))), and Q_1 = Q_2 = Q_3 iff s_1 = s_2 = s_3 = 3/4; in Kepler’s Golden Right Triangle with side lengths (1, sqrt(phi), phi), the nested radical vanishes in quadrance to give Q = (1, phi, phi^2) and exact golden spreads (s_1, s_2, s_3) = (2 - phi, phi - 1, 1) = (1/phi^2, 1/phi, 1).

Interactive GEOMETOR construction for W.11 (Isosceles, Equilateral & Kepler’s Golden Right Triangle). Hover any geometric label in the text or figure to highlight the corresponding element.

A_1 : Point A_2 : Point A_3 : Point ===== t_kepler = polygon(A_1, A_2, A_3) F_1 = foot(A_1, line(A_2, A_3)) alt = segment(A_1, F_1)

Mathematical Synthesis

In Universal Geometry, Pons Asinorum (I.5) holds for every non-null triangle over any field (Wildberger2005): \(Q_1 = Q_2 \iff s_1 = s_2\). Equilateral triangles satisfy \(Q_1 = Q_2 = Q_3 \iff s_1 = s_2 = s_3 = 3/4\), which immediately proves that no equilateral triangle can have rational coordinates in \(\mathbb{Q}^2\) because \(\mathcal{A} = 3 Q^2\) requires \(3\) to be a square in \(F\) (as it is in \(\mathbb{F}_{11}\), where \(5^2 \equiv 3\))!

Even more striking is Johannes Kepler’s Golden Right Triangle (tkepler), canonical in geometor/etc and geometor/divine. Classically burdened with the nested radical \(\sqrt{\phi}\), its quadrances are pure geometric powers of the golden ratio \((Q_3, Q_2, Q_1) = (1, \phi, \phi^2)\) satisfying \(1 + \phi = \phi^2\), while its spreads are \((1/\phi^2, 1/\phi, 1) = (2 - \phi, \phi - 1, 1)\) with altitude foot F1 dividing the hypotenuse in the exact golden section \(\phi : 1\)!

Key Algebraic Identities

\[Q_1 = Q_2 \implies s_3 = \frac{Q_3}{Q_1}\left(1 - \frac{Q_3}{4 Q_1}\right), \qquad Q_1 = Q_2 = Q_3 \iff s_1 = s_2 = s_3 = \frac{3}{4}\]

\[\text{Kepler Triangle:}\quad (Q_3, Q_2, Q_1) = (1, \phi, \phi^2), \qquad (s_3, s_2, s_1) = \left(\frac{1}{\phi^2}, \frac{1}{\phi}, 1\right) = (2 - \phi,\; \phi - 1,\; 1)\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Kepler quadrances (Q_3, Q_2, Q_1)	(1, phi, phi^2)	1 + phi = phi^2 (nested sqrt(phi) vanishes!)
Kepler spreads (s_3, s_2, s_1)	(3/2 - sqrt(5)/2, -1/2 + sqrt(5)/2, 1)	Exact golden powers (1/phi^2, 1/phi, 1)
Golden section at foot F_1	Q(F_1,A_3) / Q(A_2,F_1) = phi^2	Lengths 1 and 1/phi on hypotenuse
Equilateral spread law s_1=s_2=s_3=3/4	S(3/4, 3/4, 3/4) = 0	Verified over Z[1/2] & GF(11)

Cross-Repository Canon

Executable Demonstration	W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle — Code (Standalone model.py & w-11.svg)
Thematic Part	Part II: The Five Fundamental Laws
Euclid’s Elements	I.1, I.5, II.11, VI.30 (EuclidElements)
GEOMETOR Elements	G.2 line, G.12 rectilineal figures
GEOMETOR Fundamentals	F.18 Equilateral Triangle, F.23 Golden Triangles, F.30 Isosceles Triangles
GEOMETOR ETC & Divine	Canonical Kepler triangle (1, sqrt(phi), phi) in geometor/etc & geometor/divine

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, W.09 Law IV: The Cross Law & Quadrea-Spread Identity, W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas

Required for: W.19 Regular Stars, Golden Pentagons & Finite-Field Polygons, W.27 3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads