Posts in category '05-pentagons-polygons'

GC-05.01 · Claudius Ptolemy’s Regular Pentagon & Pentagram (Almagest I.10)

In unit circle cO with perpendicular radii OB and OC, let M bisect OB. Circle cMC cuts AB at E with \(|OB|/|EO| = \phi\) and chord \(|CE|\) equal to the side of inscribed regular pentagon CP1P2P3P4, whose pentagram diagonals P1P4, CP2, CP3 cut one another in golden chains \(|P1K2|/|K2P4| = \phi\).

GC-05.02 · Dao Thanh Oai’s Regular Pentagon Edge Perpendicular

In regular pentagon ABCDE, erect line EF perpendicular to side DE at vertex E. Line EF intersects opposite non-adjacent side AB internally at F such that \(|FB|/|AF| = |AB|/|FB| = \phi\).

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-05.04 · Tran Quang Hung’s Shared-Base Regular Pentagon & Hexagon

Erect regular hexagon ABCDEF and regular pentagon AMNPF on the same side AF. The perpendiculars MQ and PR to diagonal MP at M and P intersect upper hexagon sides BC and DE at Q and R such that \(|CQ|/|QB| = |DR|/|RE| = \phi\).

GC-05.05 · Tran Quang Hung’s Six-Pentagon Reflection Star

Reflect regular pentagon ABCDE across each of its five sides to obtain five surrounding congruent pentagons. Joining the outer lateral vertices across two intervening pentagons (such as chord S1) cuts every side of ABCDE in the golden ratio, dividing AB at G and H in a 4-point golden chain.

GC-05.06 · Tran Quang Hung’s Fivefold Wu Xing Equilateral Star

Arrange five congruent equilateral triangles OV1W1 … OV5W5 of unit side with fivefold rotational symmetry around shared vertex O. The pentagram star joining outer vertices V1 … V5 cuts the radial sides of the equilateral triangles in the golden ratio, e.g. chord V2V5 cuts OW1 at F with \(|OF|/|FW1| = \phi\).