GC-02.07 · Butterfly Chords Astride an Equilateral Triangle

Discoverer(s): Nguyen Thanh Dung | Date: 2016 | Field: \(\mathbb{Q}(\sqrt{3}, \phi)\) | Loci: 6 (Lemoine \(S = 18\))

Let I be the midpoint of radius OJ of circle cO, XT the chord perpendicular to OJ through I, and ICD an equilateral triangle symmetric about OJ with C and D on cO. If rays CI and DI meet cO again at A and B, then chords BC and AD cut XT at Y and Z with \(|YZ|/|ZT| = |YT|/|YZ| = |YZ|/|XY| = \phi\).

Given & Constructive Steps

Given: Circle cO centered at O \(= (0, 0)\) through J \(= (0, 2)\) with radius midpoint I \(= (0, 1)\).

step Draw chord XT through I perpendicular to OJ, meeting cO at X \(= (-\sqrt{3}, 1)\) and T \(= (\sqrt{3}, 1)\).

step Inscribe equilateral triangle ICD with apex I and base vertices C, D on cO.

step Extend CI and DI to meet cO at A and B, then draw butterfly chords BC and AD cutting XT at Y and Z.

Exact Symbolic Proof & Certificate

For C and D on \(x^2 + y^2 = 4\) forming equilateral ICD of side \(s\) below I \(= (0, 1)\), \((s/2)^2 + (1 - s\sqrt{3}/2)^2 = 4 \implies s^2 - \sqrt{3}\,s - 3 = 0\), giving \(s = \sqrt{3}\,\phi\) directly. By the Butterfly Theorem and symmetry across OJ, \(|IY| = |IZ| = \sqrt{3}/(2\phi)\), while \(|IT| = \sqrt{3}\). Hence \(|YZ| = \sqrt{3}/\phi\) and \(|ZT| = \sqrt{3}(1 - 1/(2\phi))\)—moreover \(|YZ|/|ZT| = |YT|/|YZ| = |YZ|/|XY| = \phi\), forming the 4-point golden chain \((X, Y, Z, T)\).

Machine-Verified Symbolic Certificate

Invariant	Exact Symbolic Value
Primary Golden Ratio	YZ \(/\) ZT \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (Y, Z, T)
Exact Segment Lengths	\(|YZ| = \frac{\sqrt{30} \sqrt{- 5 \sqrt{2} \sqrt{\sqrt{5} + 3} - \sqrt{10} \sqrt{\sqrt{5} + 3} + 6 \sqrt{5} + 18}}{10}\), \(|ZT| = \frac{\sqrt{15} \sqrt{- 10 \sqrt{2} \sqrt{\sqrt{5} + 3} - \sqrt{5} + 19 + 3 \sqrt{10} \sqrt{\sqrt{5} + 3}}}{10}\)
Rational Trigonometry Quadrances	\(Q(Y, Z) = \frac{- 15 \sqrt{2} \sqrt{\sqrt{5} + 3} - 3 \sqrt{10} \sqrt{\sqrt{5} + 3} + 18 \sqrt{5}}{10} + \frac{27}{5}\), \(Q(Z, T) = \frac{- 30 \sqrt{2} \sqrt{\sqrt{5} + 3} - 3 \sqrt{5} + 9 \sqrt{10} \sqrt{\sqrt{5} + 3}}{20} + \frac{57}{20}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)
Algebraic Number Field	\(\mathbb{Q}(\sqrt{3}, \phi)\)
Géométrographie Loci	2 given points · 5 lines · 1 circles · 1 polygons (elementary loci \(= 6\), Lemoine \(S = 18\))

Automated Divine Scan

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

• Section (X, Y, Z): long \(|YZ|^2 = \frac{- 15 \sqrt{2} \sqrt{\sqrt{5} + 3} - 3 \sqrt{10} \sqrt{\sqrt{5} + 3} + 18 \sqrt{5}}{10} + \frac{27}{5}\) vs. short \(|XY|^2 = \frac{- 30 \sqrt{2} \sqrt{\sqrt{5} + 3} - 3 \sqrt{5} + 9 \sqrt{10} \sqrt{\sqrt{5} + 3}}{20} + \frac{57}{20}\)

• Section (Y, Z, T): long \(|YZ|^2 = \frac{- 15 \sqrt{2} \sqrt{\sqrt{5} + 3} - 3 \sqrt{10} \sqrt{\sqrt{5} + 3} + 18 \sqrt{5}}{10} + \frac{27}{5}\) vs. short \(|ZT|^2 = \frac{- 30 \sqrt{2} \sqrt{\sqrt{5} + 3} - 3 \sqrt{5} + 9 \sqrt{10} \sqrt{\sqrt{5} + 3}}{20} + \frac{57}{20}\)

Connected Golden Chains:

• Chain (X — Y — Z — T) spanning 2 overlapping golden sections (progression < >)

Historical Commentary & Attribution

Discovered by Nguyen Thanh Dung (Cut-the-Knot), this synthesis of Odom’s equilateral chord and the classical Butterfly Theorem creates four collinear points \((X, Y, Z, T)\) with three overlapping golden sections on XT.

• Scholarly Citations: Cut-the-Knot: Golden Ratio Astride Equilateral Triangle, Cut-the-Knot: Golden Ratio in Geometry

• Curated Reference: Golden Ratio in a Butterfly Astride an Equilateral Triangle (curated by Alexander Bogomolny, Cut-the-Knot, © 1996–2018)

• Concepts: Butterfly Configuration · Golden Chain · Golden Section