GC-03.03 · Nguyen Thanh Dung’s 2×2 Grid Compass Arc

Discoverer(s): Nguyen Thanh Dung | Date: 2016 | Field: \(\mathbb{Q}(\phi)\) | Loci: 2 (Lemoine \(S = 6\))

On a \(2 \times 2\) square grid P1P2P3P4 with horizontal midline AB, center O, and top midpoint D, extend AB by one grid unit to C. Circle cCD centered at C through D cuts AB at M in the golden ratio: \(|MB|/|AM| = |AB|/|MB| = \phi\).
GC-03.03 — Nguyen Thanh Dung's 2×2 Grid Compass Arc cCD: y**2 + (x - 2)**2 - 5 = 0 AB: y = 0 Golden major MB = -1 + sqrt(5) Golden minor AM = 3 - sqrt(5) P1 = (-1, 1) P2 = (1, 1) P3 = (1, -1) P4 = (-1, -1) O = (0, 0) D = (0, 1) E = (0, -1) C = (2, 0) A = (-1, 0) B = (1, 0) M = (2 - sqrt(5), 0) P1 P2 P3 P4 O D E C A B M

Given & Constructive Steps

Given: A \(2 \times 2\) grid with midline endpoints A \(= (-1,0)\) and B \(= (1,0)\).

step Form the \(2 \times 2\) grid P1P2P3P4 centered at O \(= (0,0)\) with top midpoint D \(= (0,1)\).
step Mark C \(= (2,0)\) on ray AB at unit distance beyond B.
step Draw circle cCD centered at C through D, intersecting AB internally at M.

Exact Symbolic Proof & Certificate

In right triangle \(\triangle COD\), legs \(|OC| = 2\) and \(|OD| = 1\) give radius \(|CD| = \sqrt{5}\). Hence \(x_M = 2 - \sqrt{5}\), giving \(|MB| = 1 - (2 - \sqrt{5}) = \sqrt{5} - 1 = 2/\phi\) and \(|AM| = (2 - \sqrt{5}) - (-1) = 3 - \sqrt{5} = 2/\phi^2\), so \(|MB|/|AM| = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

MB \(/\) AM \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (A, M, B)

Exact Segment Lengths

\(|MB| = -1 + \sqrt{5}\), \(|AM| = 3 - \sqrt{5}\)

Rational Trigonometry Quadrances

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

Algebraic Number Field

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

Géométrographie Loci

2 given points · 1 lines · 1 circles · 1 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 (A, M, B): long \(|MB|^2 = 6 - 2 \sqrt{5}\) vs. short \(|AM|^2 = 14 - 6 \sqrt{5}\)

Historical Commentary & Attribution

Lattice coordinates supply orthogonal unit and double-unit segments without auxiliary angle bisections, making lattice-grid constructions popular in recreational and olympiad geometry.