Posts tagged 'bui-quang-tuan'

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-02.03 · Bùi Quang Tuân’s Side-and-Altitude Equilateral Triangles

Let M be the midpoint of base BC of equilateral triangle ABC, and erect a second equilateral triangle AMN on altitude AM. Then circle cBA centered at B through A cuts side MN at P in the golden ratio \(|MN|/|MP| = |MP|/|PN| = \phi\).

GC-05.03 · Prasolov & Bùi Quang Tuân’s Equi-Area Irregular Pentagon

Let ABCDE be any convex pentagon—regular or irregular—whose five peripheral triangles \(\triangle ABE, \triangle ABC, \triangle BCD, \triangle CDE, \triangle DEA\) have equal area. Then every diagonal is parallel to the opposite side and is divided by the crossing diagonals in the golden ratio, e.g. \(|BM|/|MD| = \phi\).

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-04.04 · Bùi Quang Tuân’s Van Obel Division in a Right Isosceles Triangle

In right isosceles triangle ABC with hypotenuse midpoint M and midpoint O of altitude AM, circle cOM cuts OC and OB at D and F. Cevians BD and CF concur on OM at E, dividing OM in the golden ratio: \(|OE|/|EM| = \phi\).

GC-01.07 · Bùi Quang Tuân’s Five-Step Three-Circle Secant

On line L through O, draw circle cO meeting L at A and B, circle cBA meeting L at C, and circle cCO meeting cO above L at M and cBA below L at N. Then chord MN cuts cO at P with \(|MN|/|MP| = |MP|/|PN| = \phi\).

GC-03.07 · Bùi Quang Tuân & Alexandre Borovik’s Greek Cross & Square

Inscribe a symmetrical five-square Greek cross CROSS and a concentric axis-aligned square S1S2S3S4 in the same circle cO. Along any inner arm line AC of the cross, the square edge S1S2 cuts AC at B such that \(|BC|/|AB| = \phi\), while \(|BD|/|CD| = \phi\).