Two Triangles, One Idea: The Law of Cosines as a Unifying Pedagogical Bridge Between the Isotropy of the Lorentz Factor and the Thomas Precession

A companion technical note [1] generalizes the photon-clock argument of special relativity using the law of cosines in flat (Euclidean) geometry, with full derivation and independent symbolic verification. This second note carries the same elementary construction into curved (hyperbolic) geometry and connects both to a single physical storyline: applied to a flat triangle, the law of cosines reveals the isotropy of the Lorentz factor; applied to a curved triangle in the space of relativistic rapidities, the same construction reveals the Thomas-Wigner rotation [2, 3, 4], whose infinitesimal accumulation along a curved trajectory is precisely the kinematic correction that resolves the historical factor-of-two discrepancy in the spin-orbit coupling of the hydrogen atom. The hyperbolic construction and its infinitesimal limit are derived here directly and checked against an independent numerical example; the flat-geometry companion result [1] is only summarized. No new physical result is claimed; the results assembled here are due to Einstein (1905), Varičak (1912), Thomas (1926), and Rhodes & Semon (2004).