GC-06 · Circles, Apollonian Windows, Yin–Yang & Sangaku

Infinite rational secants, tangent equal circles, Kocik’s Golden Window, Yin–Yang, and Sangaku packings

Tangent circle packings translate tangency constraints into quadratic Descartes–Pythagoras equations whose positive roots repeatedly evaluate to \(\phi\). This family catalogs Bùi Quang Tuân’s infinite Diophantine secant pencil and four-circle vesica chain, tangent circle theorems by Elliot McGucken and Thanos Kalogerakis, Jerzy Kocik’s six-circle Apollonian Golden Window, John Arioni’s Yin–Yang secant, and three equal circles inscribed in a semicircle.

Cataloged Constructions (7)

GC-06.01 · Bùi Quang Tuân’s Infinite Family of Golden Circle Secants

In unit circle cO on diameter AB, infinitely many rational-abscissa point pairs on opposite semicircles—including C1 \(= (0,1)\), D1 \(= (\tfrac{2}{3}, -\tfrac{\sqrt{5}}{3})\) and C2 \(= (-\tfrac{1}{2}, \tfrac{\sqrt{3}}{2})\), D2 \(= (\tfrac{7}{8}, -\tfrac{\sqrt{15}}{8})\)—concur on radius OB at the exact golden point M \(= (1/\phi^2, 0)\) where \(|MB|/|OM| = \phi\).

GC-06.02 · Elliot McGucken & Thanos Kalogerakis’s Equal Tangent Circles

Let equal unit circles cB and cD touch externally at C, and let AF be the diameter of cB perpendicular to line of centers BD. Secant FD cuts cD at H and I such that \(|HI|/|FH| = |FI|/|HI| = |FI|/|AF| = \phi\).

GC-06.03 · Bùi Quang Tuân’s Four Congruent Golden Circles

Let unit circles cA and cB pass through each other’s centers, intersecting at E and N, and let congruent unit circles cD and cC be externally tangent to both cA and cB. Then \((C, N, E, D)\) is a 4-point golden chain with \(|EN|/|ED| = |ND|/|EN| = \phi\).

GC-06.04 · Tran Quang Hung & Elliot McGucken’s Three Tangent Circles

Let congruent unit circles cA, cB, cC be mutually externally tangent at D, E, F, and let circle cO pass through their centers A, B, C. Line DF intersects cO at J and G to form a 4-point golden chain \(|DF|/|FG| = |DG|/|DF| = \phi\).

GC-06.05 · Jerzy Kocik’s Six-Circle Apollonian Golden Window (2010)

Place two unit-diameter circles cC1, cC2 tangent at O and two flanking congruent circles cC3, cC4 inside outer circle cO. Requiring symmetric upper and lower circles cU1, cU2 to be simultaneously tangent to all five circles uniquely forces the flanking circles to have exact radius \(\phi\), outer radius \(\phi^3\), and \(|OC4|/|C4A4| = \phi\).

GC-06.06 · John Arioni’s Golden Secant in the Yin-Yang Circle

Inscribe twin tangent circles cA and cC of radius \(\tfrac{1}{2}\) along horizontal diameter PLPR of unit circle cO, and let B be the apex of perpendicular radius OB. Secant BA cuts cA at H and I such that \(|HI|/|BH| = |BI|/|HI| = \phi\).

GC-06.07 · Three Tangent Equal Circles Inscribed in a Semicircle

Three congruent unit circles cC1, cC0, cC2 stand in a tangent row on diameter AB of enclosing circle cO centered at O, touching AB at T1, O, T2 and touching cO internally. Then cO has radius \(2\phi\) and T2 divides OB in the golden ratio: \(|OT2|/|T2B| = \phi\).