GC-02.01 · George Odom’s Equilateral Midline & Circumcircle Chord (1983)

Discoverer(s): George Odom, H. S. M. Coxeter | Date: 1983 | Field: \(\mathbb{Q}(\sqrt{3}, \phi)\) | Loci: 2 (Lemoine \(S = 6\))

Let ABC be an equilateral triangle inscribed in circle cO, and let L and M be the midpoints of sides AB and AC. Extending midline LM to meet circumcircle cO at X and Y divides the chord in the golden ratio: \(|LM|/|MY| = |LY|/|LM| = \phi\).
GC-02.01 — George Odom's Equilateral Midline & Circumcircle Chord (1983) cO: x**2 + y**2 - 4 = 0 LM: y - 1/2 = 0 Golden major LM = sqrt(3) Golden minor MY = (-sqrt(3) + sqrt(15))/2 O = (0, 0) X = (-sqrt(15)/2, 1/2) A = (0, 2) B = (-sqrt(3), -1) C = (sqrt(3), -1) L = (-sqrt(3)/2, 1/2) M = (sqrt(3)/2, 1/2) Y = (sqrt(15)/2, 1/2) O X A B C L M Y

Given & Constructive Steps

Given: Equilateral triangle ABC with vertices A \(= (0, 2)\), B \(= (-\sqrt{3}, -1)\), C \(= (\sqrt{3}, -1)\) on circumcircle cO of radius \(2\).

step Inscribe equilateral triangle ABC of side \(2\sqrt{3}\) in circle cO.
step Mark midpoints L \(= (-\tfrac{\sqrt{3}}{2}, \tfrac{1}{2})\) of AB and M \(= (\tfrac{\sqrt{3}}{2}, \tfrac{1}{2})\) of AC.
step Draw line LM parallel to BC, intersecting circumcircle cO at X \(= (-\tfrac{\sqrt{15}}{2}, \tfrac{1}{2})\) and Y \(= (\tfrac{\sqrt{15}}{2}, \tfrac{1}{2})\).

Exact Symbolic Proof & Certificate

Let \(a = |LM| = |AM| = |MC| = \sqrt{3}\) and \(b = |XL| = |MY|\). By Euclid’s Intersecting Chords Theorem (III.35) applied to chords XY and AC at M, \(|XM| \cdot |MY| = |AM| \cdot |MC|\), hence \((a + b)b = a^2\). Dividing by \(b^2\) yields \((a/b)^2 - (a/b) - 1 = 0\), proving \(|LM|/|MY| = a/b = \phi\) and forming a symmetric 4-point golden chain \((X, L, M, Y)\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

LM \(/\) MY \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (L, M, Y)

Exact Segment Lengths

\(|LM| = \sqrt{3}\), \(|MY| = \frac{- \sqrt{3} + \sqrt{15}}{2}\)

Rational Trigonometry Quadrances

\(Q(L, M) = 3\), \(Q(M, Y) = \frac{9}{2} - \frac{3 \sqrt{5}}{2}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\sqrt{3}, \phi)\)

Géométrographie Loci

3 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 (X, L, M): long \(|LM|^2 = 3\) vs. short \(|XL|^2 = \frac{9}{2} - \frac{3 \sqrt{5}}{2}\)

  • Section (L, M, Y): long \(|LM|^2 = 3\) vs. short \(|MY|^2 = \frac{9}{2} - \frac{3 \sqrt{5}}{2}\)

Connected Golden Chains:

  • Chain (X — L — M — Y) spanning 2 overlapping golden sections (progression < >)

Historical Commentary & Attribution

Discovered in the early 1980s by artist and amateur geometer George Odom at the Hudson River Psychiatric Center and communicated to H. S. M. Coxeter, who posed it as Problem E3007 in the American Mathematical Monthly (Odom & Coxeter 1983), Odom’s construction sparked an entire modern genre of equilateral-triangle golden sections.