We study the squared distance from a fixed apex to a point moving along the opposite side of a small geodesic triangle in a smooth Riemannian manifold, viewed as a \(C^2\) function of the affine cevian parameter. The first and second variations of the corresponding energy admit a single clean repackaging: an exact Volterra-type integral identity (the Jacobi-index master identity) in which the Euclidean Stewart polynomial appears as the zeroth-order term and the entire curvature content is carried by an integral of the Jacobi-index defect along the cevian variation. This places Stewart’s theorem, Apollonius’s identity, median-length formulas, and the angle-bisector ratio in a common framework and lets us ask, uniformly, which Euclidean triangle identities are stable under sectional curvature, in the sense that they either persist exactly or develop computable obstructions whose vanishing forces flatness. We obtain four structural consequences. First, a fourth-order normal-coordinate expansion of the master identity governed by the apex sectional curvature, agreeing with the constant-curvature law of cosines. Second, a curvature-scale separation between the two natural manifold analogues of the Euclidean angle-bisector foot—the geodesic-bisector foot and the Stewart-ratio foot—which coincide in flat space but differ at order \(K_A {{\,\textrm{diam}\,}}^2\) , with explicit leading coefficient. Third, asymptotic Reshetnyak-Alexandrov-type inequalities for the cevian length and bisector shift under one-sided sectional curvature bounds. Fourth, two local rigidity characterizations: vanishing of the exact Stewart correction at a single fixed cevian parameter across all small triangles forces flatness, as does coincidence of the two bisector feet along a family of non-isosceles triangle germs scanning every tangent two-plane. The master identity extends without modification to reversible Finsler manifolds; a fourth-order expansion follows unconditionally in the Berwald case.
