GC-03.02 · Tran Quang Hung’s Square Midpoint Circle Division

Discoverer(s): Tran Quang Hung | Date: 2015 | Field: \(\mathbb{Q}(\phi)\) | Loci: 4 (Lemoine \(S = 12\))

Let ABCD be a square with center F and side midpoints M, N, P, Q. The circle cKA on diameter AP intersects midline QN at X between F and N such that \(|FX|/|XN| = \phi\), while chord AX cuts MP at Z with \(|MZ|/|ZF| = \phi\).
GC-03.02 — Tran Quang Hung's Square Midpoint Circle Division cKA: y**2 + (x + 1)**2 - 5 = 0 QN: y = 0 MP: x = 0 AX: 2*x + y*(1 + sqrt(5)) - 2*sqrt(5) + 2 = 0 Golden major FX = -1 + sqrt(5) Golden minor XN = 3 - sqrt(5) C = (2, -2) D = (-2, -2) M = (0, 2) P = (0, -2) Q = (-2, 0) K = (-1, 0) Z = (0, 3 - sqrt(5)) A = (-2, 2) B = (2, 2) F = (0, 0) N = (2, 0) X = (-1 + sqrt(5), 0) C D M P Q K Z A B F N X

Given & Constructive Steps

Given: Square ABCD of side \(4\) centered at F \(= (0,0)\) with vertices \((\pm 2, \pm 2)\).

step Mark side midpoints M \(= (0,2)\), N \(= (2,0)\), P \(= (0,-2)\), Q \(= (-2,0)\) and center F \(= (0,0)\).
step Bisect AP at K \(= (-1,0)\) and draw circle cKA of radius \(\sqrt{5}\) on diameter AP.
step Mark X where cKA crosses QN between F and N, and Z where AX crosses MP.

Exact Symbolic Proof & Certificate

Circle cKA has equation \((x + 1)^2 + y^2 = 5\). Along QN (\(y = 0\)), the positive root is \(x_X = \sqrt{5} - 1 = 2/\phi\). Since \(x_N = 2\), \(|XN| = 3 - \sqrt{5} = 2/\phi^2\), whence \(|FX|/|XN| = (2/\phi)/(2/\phi^2) = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

FX \(/\) XN \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (F, X, N)

Exact Segment Lengths

\(|FX| = -1 + \sqrt{5}\), \(|XN| = 3 - \sqrt{5}\)

Rational Trigonometry Quadrances

\(Q(F, X) = 6 - 2 \sqrt{5}\), \(Q(X, N) = 14 - 6 \sqrt{5}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

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

Géométrographie Loci

2 given points · 3 lines · 1 circles · 1 polygons (elementary loci \(= 4\), Lemoine \(S = 12\))

Automated Divine Scan

geometor.divine.golden detected 4 exact collinear golden section(s) and 1 connected golden chain(s) in this model:

  • Section (Q, F, X): long \(|QF|^2 = 4\) vs. short \(|FX|^2 = 6 - 2 \sqrt{5}\)

  • Section (F, X, N): long \(|FX|^2 = 6 - 2 \sqrt{5}\) vs. short \(|XN|^2 = 14 - 6 \sqrt{5}\)

  • Section (F, Z, M): long \(|ZM|^2 = 6 - 2 \sqrt{5}\) vs. short \(|FZ|^2 = 14 - 6 \sqrt{5}\)

  • Section (A, Z, X): long \(|AZ|^2 = 10 - 2 \sqrt{5}\) vs. short \(|ZX|^2 = 20 - 8 \sqrt{5}\)

Connected Golden Chains:

  • Chain (Q — F — X — N) spanning 2 overlapping golden sections (progression > >)

Historical Commentary & Attribution

Right angle \(\angle AXP = 90^\circ\) (Thales’ theorem on diameter AP) also makes \(\triangle PFX \sim \triangle AFZ\), transferring the golden division from horizontal half-midline FN directly onto vertical half-midline FM.