GC-04.01 · H. E. Huntley’s Golden Ratio in the 3-4-5 Right Triangle (1970)

Discoverer(s): H. E. Huntley, Tran Quang Hung | Date: 1970 | Field: \(\mathbb{Q}(\phi)\) | Loci: 2 (Lemoine \(S = 6\))

In a \(3\text{-}4\text{-}5\) right triangle ABC (\(|BC|=3, |AC|=4, |AB|=5\)), let the internal bisector of \(\angle ABC\) meet leg AC at O. Circle cOC centered at O through C intersects line BO at P and Q such that \(|PQ|/|BP| = |BQ|/|PQ| = \phi\).
GC-04.01 — H. E. Huntley's Golden Ratio in the 3-4-5 Right Triangle (1970) cOC: x**2 + (y - 3/2)**2 - 9/4 = 0 BO: -3*x/2 - 3*y + 9/2 = 0 Golden major PQ = 3 Golden minor BP = -3/2 + 3*sqrt(5)/2 A = (0, 4) O = (0, 3/2) T = (6/5, 12/5) C = (0, 0) B = (3, 0) P = (3*sqrt(5)/5, 3/2 - 3*sqrt(5)/10) Q = (-3*sqrt(5)/5, 3*sqrt(5)/10 + 3/2) A O T C B P Q

Given & Constructive Steps

Given: Right triangle ABC with right angle at C \(= (0,0)\), B \(= (3,0)\), and A \(= (0,4)\).

step Bisect \(\angle ABC\) to meet leg AC at O \(= (0, \tfrac{3}{2})\).
step Draw circle cOC of radius \(\tfrac{3}{2}\) centered at O through C (tangent to both BC at C and hypotenuse AB at T).
step Extend bisector BO across cOC to obtain secant intersections P and Q.

Exact Symbolic Proof & Certificate

By the angle-bisector theorem, \(|CO|/|OA| = |BC|/|AB| = 3/5\), giving \(|CO| = 3/2\) and diameter \(|PQ| = 3 = |BC|\). By Euclid III.36 (tangent-secant power from B to cOC), \(|BC|^2 = |BP| \cdot |BQ| = |BP|(|BP| + |PQ|)\). Substituting \(|BC| = |PQ|\) gives \(|PQ|^2 = |BP|^2 + |BP|\cdot|PQ|\), whence \(|PQ|/|BP| = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

PQ \(/\) BP \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (B, P, Q)

Exact Segment Lengths

\(|PQ| = 3\), \(|BP| = - \frac{3}{2} + \frac{3 \sqrt{5}}{2}\)

Rational Trigonometry Quadrances

\(Q(P, Q) = 9\), \(Q(B, P) = \frac{27}{2} - \frac{9 \sqrt{5}}{2}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

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 (Q, P, B): long \(|QP|^2 = 9\) vs. short \(|PB|^2 = \frac{27}{2} - \frac{9 \sqrt{5}}{2}\)

Historical Commentary & Attribution

Featured in H. E. Huntley’s The Divine Proportion (1970), this theorem links the oldest Pythagorean triple \((3,4,5)\) directly to extreme and mean ratio.