GC-07.01 · Tran Quang Hung’s 2:1 Rhombus & Incircle Diagonal Division

Discoverer(s): Tran Quang Hung | Date: 2015 | Field: \(\mathbb{Q}(\phi)\) | Loci: 2 (Lemoine \(S = 6\))

Let ABCD be a rhombus whose diagonals satisfy \(|DB| = 2|AC|\), and let cO be its inscribed circle. Circle cO intersects long diagonal DB at E and F to form a 4-point golden chain: \(|EF|/|FB| = |EB|/|EF| = \phi\).
GC-07.01 — Tran Quang Hung's 2:1 Rhombus & Incircle Diagonal Division cO: x**2 + y**2 - 4 = 0 DB: y = 0 Golden major EF = 4 Golden minor FB = -2 + 2*sqrt(5) A = (0, sqrt(5)) C = (0, -sqrt(5)) D = (-2*sqrt(5), 0) O = (0, 0) B = (2*sqrt(5), 0) F = (2, 0) E = (-2, 0) A C D O B F E

Given & Constructive Steps

Given: Rhombus ABCD centered at O \(= (0,0)\) with B \(= (2\sqrt{5}, 0)\) and A \(= (0, \sqrt{5})\).

step Construct rhombus ABCD of side \(5\) with semi-diagonals \(|OB| = 2\sqrt{5}\) and \(|OA| = \sqrt{5}\).
step Inscribe circle cO of radius \(r = 2\) tangent to all four sides of ABCD.
step Mark intersections E \(= (-2,0)\) and F \(= (2,0)\) of cO with diagonal DB.

Exact Symbolic Proof & Certificate

Right triangle \(\triangle AOB\) has legs \(\sqrt{5}\) and \(2\sqrt{5}\), hypotenuse \(|AB| = 5\), and altitude (inradius) \(r = \frac{\sqrt{5}\cdot 2\sqrt{5}}{5} = 2\). Thus diameter \(|EF| = 4\), while \(|FB| = 2\sqrt{5} - 2 = 2(\sqrt{5}-1) = 4/\phi\), giving \(|EF|/|FB| = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

EF \(/\) FB \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (E, F, B)

Exact Segment Lengths

\(|EF| = 4\), \(|FB| = -2 + 2 \sqrt{5}\)

Rational Trigonometry Quadrances

\(Q(E, F) = 16\), \(Q(F, B) = 24 - 8 \sqrt{5}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\phi)\)

Géométrographie Loci

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

Automated Divine Scan

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

  • Section (D, E, F): long \(|EF|^2 = 16\) vs. short \(|DE|^2 = 24 - 8 \sqrt{5}\)

  • Section (E, F, B): long \(|EF|^2 = 16\) vs. short \(|FB|^2 = 24 - 8 \sqrt{5}\)

Connected Golden Chains:

  • Chain (D — E — F — B) spanning 2 overlapping golden sections (progression < >)

Historical Commentary & Attribution

Scaling the semi-diagonals by \(\sqrt{5}\) makes both the rhombus side (\(5\)) and the inradius (\(2\)) pure rational integers.