A Chebyshev Criterion for Angle Multiples in a Triangle

The familiar condition A = 2B if and only if a^2 - b^2 = 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 = (a^2-b^2)/(bc) and u = (a^2+c^2-b^2)/(ac) = 2 cos B. For an integer k >= 2, if A = kB, then M = Pₖ₋₁(u), where P_0 = 0, P_1 = 1, and Pₙ₊₁ = uP_n - Pₙ₋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, a^2-b^2 = bc and, in the sector 0 < B < pi/4, (a+b)(a-b)^2 = bc^2.