A Chebyshev Criterion for Angle Multiples in a Triangle

The familiar condition A = 2B if and only if a² - b² = bc for a triangle is often treated as an isolated double-angle identity. We show that it is the first case of a uniform Chebyshev side equation. Let the angles of a nondegenerate triangle ABC be opposite the sides a, b, c, and set M = (a²-b²) / (bc) and u = (a²+c²-b²) / (ac) = 2 cos B. For an integer k >= 2, if A = kB, then M = P₊-₁ (u), where P₀ = 0, P₁ = 1, and P₍+₁ = uPₙ - P₍-₁ for n >= 1. Conversely, in the natural sector 0 < B < pi/ (k+1), this same side equation is equivalent to A = kB or to the sine-reflected branch C = (k-3) B; for k = 2 and k = 3 the reflected branch is not a nondegenerate triangle. The cases k = 2 and k = 3 give, respectively, a²-b² = bc and, in the sector 0 < B < pi/4, (a+b) (a-b) ² = bc².