GC-06.04 · Tran Quang Hung & Elliot McGucken’s Three Tangent Circles

Discoverer(s): Tran Quang Hung, Elliot McGucken | Date: 2016 | Field: \(\mathbb{Q}(\sqrt{3}, \phi)\) | Loci: 5 (Lemoine \(S = 15\))

Let congruent unit circles cA, cB, cC be mutually externally tangent at D, E, F, and let circle cO pass through their centers A, B, C. Line DF intersects cO at J and G to form a 4-point golden chain \(|DF|/|FG| = |DG|/|DF| = \phi\).
GC-06.04 — Tran Quang Hung & Elliot McGucken's Three Tangent Circles cA: x**2 + (y - 2*sqrt(3)/3)**2 - 1 = 0 cB: (x + 1)**2 + (y + sqrt(3)/3)**2 - 1 = 0 cC: (x - 1)**2 + (y + sqrt(3)/3)**2 - 1 = 0 cO: x**2 + y**2 - 4/3 = 0 DF: -sqrt(3)*x/2 - y/2 - sqrt(3)/6 = 0 Golden major DF = 1 Golden minor FG = -1/2 + sqrt(5)/2 B = (-1, -sqrt(3)/3) C = (1, -sqrt(3)/3) E = (1/2, sqrt(3)/6) J = (-1/4 + sqrt(5)/4, -sqrt(15)/4 - sqrt(3)/12) O = (0, 0) A = (0, 2*sqrt(3)/3) D = (0, -sqrt(3)/3) F = (-1/2, sqrt(3)/6) G = (-sqrt(5)/4 - 1/4, -sqrt(3)/12 + sqrt(15)/4) B C E J O A D F G

Given & Constructive Steps

Given: Equilateral center triangle ABC of side \(2\) centered at O \(= (0,0)\) with A \(= (0, \tfrac{2\sqrt{3}}{3})\).

step Draw mutually tangent unit circles cA, cB, cC touching at midpoints D, E, F of ABC.
step Draw circumcircle cO of radius \(\tfrac{2\sqrt{3}}{3}\) through centers A, B, C.
step Extend midline DF across cO to obtain intersections G and J.

Exact Symbolic Proof & Certificate

Because D, E, F are the midpoints of equilateral ABC inscribed in cO, chord JG along midline DF reproduces George Odom’s configuration (GC-02.01 · George Odom’s Equilateral Midline & Circumcircle Chord (1983)) inside the three-circle packing: \(|DF| = 1\) and \(|FG| = |DJ| = \frac{\sqrt{5}-1}{2} = 1/\phi\), hence \(|DF|/|FG| = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

DF \(/\) FG \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (D, F, G)

Exact Segment Lengths

\(|DF| = 1\), \(|FG| = - \frac{1}{2} + \frac{\sqrt{5}}{2}\)

Rational Trigonometry Quadrances

\(Q(D, F) = 1\), \(Q(F, G) = \frac{3}{2} - \frac{\sqrt{5}}{2}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

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

Géométrographie Loci

2 given points · 1 lines · 4 circles · 1 polygons (elementary loci \(= 5\), Lemoine \(S = 15\))

Automated Divine Scan

geometor.divine.golden detected 2 exact collinear golden section(s) and 1 connected golden chain(s) in this model:

  • Section (G, F, D): long \(|FD|^2 = 1\) vs. short \(|GF|^2 = \frac{3}{2} - \frac{\sqrt{5}}{2}\)

  • Section (F, D, J): long \(|FD|^2 = 1\) vs. short \(|DJ|^2 = \frac{3}{2} - \frac{\sqrt{5}}{2}\)

Connected Golden Chains:

  • Chain (G — F — D — J) spanning 2 overlapping golden sections (progression < >)

Historical Commentary & Attribution

Furthermore, extending DF across cC adds another unit chord segment beyond D, weaving six circle intersections along a single line.