GC-03.01 · Tran Quang Hung’s Square Inscribed in a Right Isosceles Semicircle

Discoverer(s): Tran Quang Hung | Date: 2014 | Field: \(\mathbb{Q}(\phi)\) | Loci: 3 (Lemoine \(S = 9\))

Let ABC be a right isosceles triangle inscribed in a semicircle of cO on diameter AB, and let square DEGF sit on AB with vertices D and E on legs AC and BC. Extending top side DE to meet circle cO at Q and P divides DP in the golden ratio: \(|DE|/|EP| = |DP|/|DE| = \phi\).
GC-03.01 — Tran Quang Hung's Square Inscribed in a Right Isosceles Semicircle cO: x**2 + y**2 - 9 = 0 AB: y = 0 DE: y - 2 = 0 Golden major DE = 2 Golden minor EP = -1 + sqrt(5) C = (0, 3) G = (1, 0) F = (-1, 0) Q = (-sqrt(5), 2) O = (0, 0) A = (-3, 0) B = (3, 0) D = (-1, 2) E = (1, 2) P = (sqrt(5), 2) C G F Q O A B D E P

Given & Constructive Steps

Given: Circle cO centered at O \(= (0,0)\) of radius \(3\) with diameter A \(= (-3,0)\), B \(= (3,0)\) and apex C \(= (0,3)\).

step Inscribe right isosceles triangle ABC in the upper semicircle on diameter AB.
step Inscribe square DEGF of side \(2\) with F \(= (-1,0)\), G \(= (1,0)\) on AB and D \(= (-1,2)\), E \(= (1,2)\) on AC, BC.
step Extend line DE horizontally to meet circle cO at Q \(= (-\sqrt{5}, 2)\) and P \(= (\sqrt{5}, 2)\).

Exact Symbolic Proof & Certificate

At \(x = \pm 1\), legs AC (\(y = x + 3\)) and BC (\(y = -x + 3\)) have height \(y = 2\), matching the horizontal side \(|DE| = 2\) of inscribed square DEGF. On circle \(x^2 + y^2 = 9\), setting \(y = 2\) yields \(x = \pm\sqrt{5}\), so \(|EP| = \sqrt{5} - 1 = 2/\phi\) and \(|DE|/|EP| = 2 / (2/\phi) = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

DE \(/\) EP \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (D, E, P)

Exact Segment Lengths

\(|DE| = 2\), \(|EP| = -1 + \sqrt{5}\)

Rational Trigonometry Quadrances

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

Algebraic Number Field

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

Géométrographie Loci

3 given points · 2 lines · 1 circles · 2 polygons (elementary loci \(= 3\), Lemoine \(S = 9\))

Automated Divine Scan

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

  • Section (Q, D, E): long \(|DE|^2 = 4\) vs. short \(|QD|^2 = 6 - 2 \sqrt{5}\)

  • Section (D, E, P): long \(|DE|^2 = 4\) vs. short \(|EP|^2 = 6 - 2 \sqrt{5}\)

Connected Golden Chains:

  • Chain (Q — D — E — P) spanning 2 overlapping golden sections (progression < >)

Historical Commentary & Attribution

Because \((Q, D, E, P)\) is symmetric across the vertical axis, it forms a complete 4-point golden chain with lengths \(2/\phi, 2, 2/\phi\).