GC-02.02 · Michel Bataille’s Equilateral Triangle & External Square

Discoverer(s): Michel Bataille | Date: 2011 | Field: \(\mathbb{Q}(\sqrt{3}, \phi)\) | Loci: 2 (Lemoine \(S = 6\))

Erect equilateral triangle ABC and external square BCDE on opposite sides of shared base BC. Circle cCE centered at C through opposite square vertex E cuts ray AB beyond B at F so that \(|AF|/|AB| = |AB|/|BF| = \phi\).
GC-02.02 — Michel Bataille's Equilateral Triangle & External Square cCE: y**2 + (x - 2)**2 - 8 = 0 AB: sqrt(3)*x - y = 0 Golden major AB = 2 Golden minor BF = -1 + sqrt(5) D = (2, -2) E = (0, -2) C = (2, 0) B = (0, 0) A = (1, sqrt(3)) F = (1/2 - sqrt(5)/2, -sqrt(15)/2 + sqrt(3)/2) D E C B A F

Given & Constructive Steps

Given: Shared base BC from B \(= (0, 0)\) to C \(= (2, 0)\) of length \(2\).

step Construct equilateral triangle ABC above BC with apex A \(= (1, \sqrt{3})\).
step Construct square BCDE below BC with D \(= (2, -2)\) and E \(= (0, -2)\).
step Draw circle cCE centered at C through E (radius \(2\sqrt{2}\)), intersecting line AB beyond B at F.

Exact Symbolic Proof & Certificate

Diagonal CE of square BCDE has quadrance \(Q(C, E) = 2^2 + 2^2 = 8\). Along ray AB extended beyond B by distance \(s = |BF|\), \(\angle CBF = 120^\circ\), so the Law of Cosines in \(\triangle CBF\) gives \(s^2 + 2s + 4 = 8\), or \(s^2 + 2s - 4 = 0\). Thus \(|BF| = \sqrt{5} - 1 = 2/\phi\) and \(|AF| = 2 + (\sqrt{5} - 1) = \sqrt{5} + 1 = 2\phi\), proving \(|AF|/|AB| = |AB|/|BF| = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

AB \(/\) BF \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (A, B, F)

Exact Segment Lengths

\(|AB| = 2\), \(|BF| = -1 + \sqrt{5}\)

Rational Trigonometry Quadrances

\(Q(A, B) = 4\), \(Q(B, F) = 6 - 2 \sqrt{5}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\sqrt{3}, \phi)\)

Géométrographie Loci

2 given points · 1 lines · 1 circles · 2 polygons (elementary loci \(= 2\), Lemoine \(S = 6\))

Automated Divine Scan

geometor.divine.golden detected 1 exact collinear golden section(s) and 0 connected golden chain(s) in this model:

  • Section (F, B, A): long \(|BA|^2 = 4\) vs. short \(|FB|^2 = 6 - 2 \sqrt{5}\)

Historical Commentary & Attribution

Published by Michel Bataille (Bataille 2011, Cut-the-Knot), this construction bridges the regular \(3\)-gon and \(4\)-gon directly to \(\phi\).