This paper establishes the precise geometric correlation between the family of metallic ratios and the different families of primitive Pythagorean triples. It demonstrates that every metallic mean δₙ (n ≥ 5) appears exactly as the cotangent of one-quarter of the smaller acute angle in a specific primitive Pythagorean triple, with explicit generating formulae provided for the Pythagoras, Plato and Socrates families.Special emphasis is placed on the first metallic mean — the Golden Ratio (δ₁). It is geometrically substantiated in the classic 3-4-5 Pythagorean triple as δ₁ = cot((θ + 90°)/4), where θ is the smaller acute angle. Remarkably, the Golden Ratio is also precisely recovered through the summation of the larger acute angles from carefully certain pairs of Pythagorean triples. For example, δ₁ = cot((φ₁ + φ₂)/4), where φ₁ and φ₂ are the larger acute angles of the (5,12,13) and (33,56,65) triples (and similarly for many other documented pairs such as (3,4,5) & (7,24,25), (8,15,17) & (36,77,85), etc.).The paper further reveals trigonometric expressions for all metallic means, “triad” algebraic relationships between different metallic ratios, and geometric substantiation in non-integer right triangles. These results show that right-angled triangles and primitive Pythagorean triples are the most prototypical geometric embodiments of the entire metallic-means family.Keywords: Metallic Means, Metallic Ratios, Golden Ratio, Pythagorean Triples, 3-4-5 Triangle, Right Triangles, Trigonometric Identities
Chetansing Rajput (Mon,) studied this question.