What question did this study set out to answer?

To derive the golden triangle's side ratios and their connection to the golden ratio phi using geometric principles.

July 26, 2026Open Access

Golden Triangle Derivation of φ via Angle Bisector and Similar Triangles — E8 Intelligence Research

Key Points

To derive the golden triangle's side ratios and their connection to the golden ratio phi using geometric principles.
Constructed angle bisector of a 36° base angle in a 108-36-36 triangle to create similar isosceles triangles.
Developed a mathematical proportion resulting in the equation for phi.
Analyzed properties of both the golden triangle and related geometric constructions.
Derived phi as (1+√5)/2 ≈ 1.618 from the triangle's side ratios.
Identified that sin 18° ≈ 0.309 and cos 36° ≈ 0.809 correlate with phi.
Demonstrated that these triangles tile a regular pentagon, revealing deeper geometric symmetries.

Abstract

FINDING: Golden triangle (108-36-36) side ratio derivation via similar triangles yields φ exactly. | MATH: In a 108-36-36 triangle with sides (1,1,φ), the ratio of leg to base is φ:1. Derivation: construct angle bisector of 36° base angle, creating similar isosceles triangles; set up proportion (1/φ) = (φ-1)/1 → φ² - φ - 1 = 0 → φ = (1+√5)/2 ≈ 1.618. Also yields sin 18° = (√5-1)/4 ≈ 0.309, cos 36° = φ/2 ≈ 0.809. | CONNECTION: φ appears directly; related ratios: 1/φ ≈ 0.618, 1/φ² ≈ 0.382, φ² ≈ 2.618. The 36-72-72 triangle (golden gnomon) has side ratios 1:1:1/φ. Both triangles tile the regular pentagon, whose diagonals intersect in φ ratios. | DEPTH: 7 — fundamental geometric construction linking φ to pentagonal symmetry, which underlies icosahedral/dodecahedral crystallographic point groups (5-fold axes) and quasicrystal diffraction patterns. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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