Golden Triangle Side Ratio Derived from Golden Ratio via Angle Bisector Similarity — E8 Intelligence Research

FINDING: The golden triangle (36-72-72 or 108-36-36) derives its side ratio directly from the golden ratio φ via similarity of isosceles triangles formed by an angle bisector. MATH: - For a 36-72-72 triangle with sides (base, equal legs) = (1, φ), the angle bisector of a base angle creates a smaller similar triangle, yielding φ = (1 + √5)/2 ≈ 1.618. - For a 108-36-36 triangle (apex 108°, base angles 36°), the side ratio (equal leg : base) = φ : 1, i.e., leg = φ × base. - Key identity: 2 cos(36°) = φ, and 2 cos(72°) = 1/φ = φ − 1 ≈ 0.618. - Sine/cosine exact values: sin(18°) = (√5 − 1)/4, cos(36°) = φ/2, etc. CONNECTION: - The 36-72-72 triangle is the face of the regular pentagon; its diagonals intersect in φ ratios. - The 108-36-36 triangle appears in pentagram star points; side ratios are φ : 1 : φ. - Both triangles are directly linked to the golden ratio φ (1.618) and its reciprocal 0.618, and to the complementary ratio 0.382 ( = 1 − 1/φ). - These ratios are fundame Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com