F.23 Golden Triangles

The 72-72-36 Sublime Triangle, 5th Roots of Unity, and Normalized Stewart’s Theorem

In the acute golden isosceles triangle ABC with base AB = 1, legs AC = BC = phi, and angles 72-72-36 degrees, bisecting base angle B meets AC at D such that CD = 1, DA = phi - 1 = 1/phi, and BD = AB = 1, partitioning ABC into a smaller golden triangle BCD and a golden gnomon ABD.

Interactive GEOMETOR construction for F.23 (Golden Triangles). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point ===== C = golden_apex(A, B) ABC = polygon(A, B, C) D = golden_section(A, C) BD = segment(B, D)

Mathematical Synthesis

Constructed in EuclidElements IV.10 (G.25 in geometor/elements), the acute golden triangle ABC has leg-to-base ratio equal to the golden ratio \(\phi = \frac{1+\sqrt{5}}{2}\). In Topic 23 (SeeligNotebooks), Seelig computes the altitude \(CM = \sqrt{\phi^2 - 1/4} = \frac{\sqrt{\phi + 3/4}}{\phi}\) and segment \(AD = 2\sin 36^\circ = \sqrt{3 - \phi} \approx 1.1756\) when D divides leg \(BC\) into \(\phi - 1\) and \(1\).

In Folio 2 (FWS 12/21/2014), he constructs the nested ‘GOLDEN IDENTITY TRIANGLES’ (right triangles sharing unit altitude with base angles \(72^\circ, 54^\circ, 36^\circ, 18^\circ\)) and tabulates the 5th roots of unity \(e^{i 2\pi n/5}\). In Folio 3 (FWS 11/18/2013), he proves the normalized form of Stewart's theorem \(f^2 = k a^2 + k' b^2 - k k' c^2\) (\(k + k' = 1\)).

Key Algebraic Identities

\[\phi = \frac{1 + \sqrt{5}}{2}, \qquad \cos 36^\circ = \frac{\phi}{2}, \qquad \cos 72^\circ = \frac{\phi - 1}{2} = \frac{1}{2\phi}, \qquad 2\sin 36^\circ = \sqrt{3 - \phi}\]

\[f^2 = k a^2 + k' b^2 - k k' c^2 \qquad (k + k' = 1)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.23 Golden Triangles — Folios (Folios 0132–0134, 3 scanned manuscript pages in resources/seelig/23_golden-triangles)
Euclid’s Elements	II.11, IV.10, IV.11, VI.30 (EuclidElements)
GEOMETOR Elements (G-Index)	G.10 circle, G.25 base angles of an isosceles triangle
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Golden Triangle, Polygon Table (Spreadsheets)

Computational Workbook Verification

The following worksheets from seelig’s Triangles and Circles.ods workbook verify the numerical and symbolic formulas for this topic on the canonical \((a, b, c) = (5, 6, 7)\) triangle:

Worksheet: Golden Triangle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
F11	F11	0.09549	=F13^2
F12	F12	0.19098	=1-[.$F$36]/2
F13	F13	0.30902	=1/(2*[.F$36])
F14	F14	0.32492	=TAN(RADIANS(18))
F15	|IM|	0.36327	=F35/F42
F16	|GM|	0.51295	=F35/3
F17	F17	0.52786	=(F36-0.5)/(F36+0.5)
F18	F18	0.58779	=F31/2

Cell Excerpt

The Golden Triangle	—	—	—	—	—
Isosceles triangle with equal sides	—	—	—	—	—
Unity base length, vertex angle 36°,	—	—	—	—	—
Fred W. Seelig	—	—	—	—	—
Mar. 22, 2015	—	—	Created	—	—
Mar. 31, 2015	—	—	Last Modified	—	—

Constructive Dependency Graph

Prerequisites: F.7 Ceva and Menelaus, F.18 Equilateral Triangle

Required for: F.40 Regular Polygons and Nested Radicals