π-thagoras : Pi-thagoras

A universal geometric bridge from any primitive Pythagorean triple to π/2: continued-fraction convergents of a carefully chosen target ratio generate partner triples whose angles make cot((α0 + αk)/4) converge rapidly to π/2. For any fixed primitive Pythagorean triple (a0, b0, c0) with larger acute angle α0, we prove that there exists an infinite sequence of partner primitive Pythagorean triples (ak, bk, ck) such that cot((α0 + αk)/4) → π/2 (k → ∞), where αk is derived from the Euclid parameters of the k-th partner triple. The partner triples are the continued-fraction convergents (with a parity correction using semiconvergents) of an explicit target ratio t = tan(α∗/2), where α∗ = 4 arccot(π/2) − α0 is a fixed target angle determined by α0. We give a complete proof of convergence, an explicit error bound, and verify the construction numerically to at least 8 significant digits of π/2 for eighteen fixed triples.