GC-07 · Rhombi, Trapezoids, Mixtilinear Circles & Dissections

2:1 and golden rhombi, 1:3 area-bisecting trapezoids, mixtilinear circles, and Hooper–Curry resolution

Rounding out the catalog, quadrilaterals and curvilinear touchings demonstrate how quadratic means, tangency altitudes, and Cassini’s Fibonacci identity converge on \(\phi\). Entries span Tran Quang Hung’s \(2:1\) rhombus incircle, \(60^\circ\) half-hexagon trapezoid, and equilateral mixtilinear circle; John Molokach’s golden rhombus from shifted \(1 \times 3\) rectangles; Jingcheng Tong’s root-mean-square trapezoid bisector; and the exact \(\phi\)-power resolution of the William Hooper–Paul Curry dissection paradox.

Cataloged Constructions (6)

GC-07.01 · Tran Quang Hung’s 2:1 Rhombus & Incircle Diagonal Division

Let ABCD be a rhombus whose diagonals satisfy \(|DB| = 2|AC|\), and let cO be its inscribed circle. Circle cO intersects long diagonal DB at E and F to form a 4-point golden chain: \(|EF|/|FB| = |EB|/|EF| = \phi\).

GC-07.02 · John Molokach’s Golden Rhombus from Shifted 1×3 Rectangles

Stack two \(3 \times 1\) rectangles RECT1 and RECT2 shifted by one unit around center O, draw unit circle cO, and draw the two tangents to cO perpendicular to outer diagonal R1R6. Together with the horizontal outer edges, they bound a golden rhombus ABCD (\(|AC|/|BD| = \phi\)) with \(|T1K|/|KA| = \phi\).

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

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-07.04 · Tran Quang Hung’s 60° Isosceles Trapezoid Diagonal Division

In half-hexagon isosceles trapezoid ABCD (\(|DA| = |AB| = |BC|\) and \(\angle C = \angle D = 60^\circ\)), reflect B across C to K. Circle cKB centered at K through B cuts diagonal AC internally at M such that \(|CM|/|MA| = |CA|/|CM| = \phi\).

GC-07.05 · Tran Quang Hung’s Mixtilinear Circle in an Equilateral Triangle

In equilateral triangle ABC with circumcenter K and circumcircle cK, inscribe the A-mixtilinear circle cF tangent to sides AB, AC and internally to cK. The line through K parallel to AB meets BC at O and cF at I and J such that \(|KO|/|OJ| = \phi\).

GC-07.06 · Exact φ-Limit Resolution of the Hooper–Curry Fibonacci Dissection

Replace integer Fibonacci lengths \((3, 5, 8, 13)\) in the \(8 \times 8 = 64\) vs. \(5 \times 13 = 65\) dissection fallacy by exact powers \((1, \phi, \phi^2, \phi^3)\). In rectangle ABCD of width \(\phi^3\) and height \(\phi\), internal vertices Q \(= (\phi, 1/\phi)\) and P \(= (\phi^2, 1)\) lie identically on diagonal AC, forming the 4-point golden chain \(|AQ|/|QP| = |AP|/|AQ| = \phi\).