GC-07.03 · Jingcheng Tong’s Area-Bisecting Chord in a 1:3 Trapezoid

Discoverer(s): Jingcheng Tong | Date: 2005 | Field: \(\mathbb{Q}(\phi)\) | Loci: 3 (Lemoine \(S = 9\))

Let ABCD be a trapezoid with parallel bases AD and BC satisfying \(|BC| = 3|AD|\). The chord MN parallel to the bases that bisects the area of ABCD divides both non-parallel legs AB and DC in the golden ratio: \(|AM|/|MB| = |DN|/|NC| = \phi\).
GC-07.03 — Jingcheng Tong's Area-Bisecting Chord in a 1:3 Trapezoid AB: 2*x - 2*y + 6 = 0 DC: 2*x + 2*y - 6 = 0 MN: y - 3 + sqrt(5) = 0 Golden major AM = -sqrt(2) + sqrt(10) Golden minor MB = -sqrt(10) + 3*sqrt(2) D = (1, 2) C = (3, 0) N = (sqrt(5), 3 - sqrt(5)) A = (-1, 2) B = (-3, 0) M = (-sqrt(5), 3 - sqrt(5)) D C N A B M

Given & Constructive Steps

Given: Trapezoid ABCD with top base A \(= (-1,2)\) to D \(= (1,2)\) (\(a = 2\)) and bottom base B \(= (-3,0)\) to C \(= (3,0)\) (\(b = 6\)).

step Construct trapezoid ABCD with parallel bases of lengths \(a = 2\) and \(b = 6\).
step Compute the root-mean-square chord length \(c = \sqrt{(a^2 + b^2)/2} = 2\sqrt{5}\).
step Draw parallel chord MN at height \(y = 3 - \sqrt{5}\) where horizontal width equals \(2\sqrt{5}\).

Exact Symbolic Proof & Certificate

Any parallel chord of length \(c\) divides legs AB and DC in the ratio \(|AM|/|MB| = (c - a)/(b - c)\), and bisects the trapezoid’s area precisely when \(c^2 = (a^2 + b^2)/2\). For \(b = 3a\), \(c = \sqrt{5}\,a\), giving \(|AM|/|MB| = (\sqrt{5} - 1)/(3 - \sqrt{5}) = \phi\)—independent of leg slopes or height!

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

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

Exact Segment Lengths

\(|AM| = - \sqrt{2} + \sqrt{10}\), \(|MB| = - \sqrt{10} + 3 \sqrt{2}\)

Rational Trigonometry Quadrances

\(Q(A, M) = 12 - 4 \sqrt{5}\), \(Q(M, B) = 28 - 12 \sqrt{5}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

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

Géométrographie Loci

2 given points · 3 lines · 0 circles · 1 polygons (elementary loci \(= 3\), Lemoine \(S = 9\))

Automated Divine Scan

geometor.divine.golden detected 2 exact collinear golden section(s) and 0 connected golden chain(s) in this model:

  • Section (B, M, A): long \(|MA|^2 = 12 - 4 \sqrt{5}\) vs. short \(|BM|^2 = 28 - 12 \sqrt{5}\)

  • Section (D, N, C): long \(|DN|^2 = 12 - 4 \sqrt{5}\) vs. short \(|NC|^2 = 28 - 12 \sqrt{5}\)

Historical Commentary & Attribution

Because area bisection depends only on the base ratio \(b/a = 3\), this theorem holds for every trapezoid (isosceles or scalene) whose parallel bases stand in ratio \(1 : 3\).