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

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