Similar triangles inscribed in a Jordan curve : At most two exceptional vertices for every nondegenerate shape

We prove that, for every nondegenerate triangle shape and every planar Jordan curve, all but at most two points of the curve are vertices of an inscribed triangle similar to that shape. Both orientation-preserving and orientation-reversing similarities are allowed, and the vertex role is not prescribed. The bound is sharp for every shape. The proof uses integer winding numbers associated individually with similarities centered at an exceptional point. The ordering of branches in the logarithmic covering forces their nonzero values to have a common sign, without any assumption of radial monotonicity. Relay identities between three centers force these winding numbers to vanish once nested copies have been excluded. Pair and triple identities then yield an integer contradiction. All local arguments apply to continuous Jordan curves, without assuming one-sided tangents or rectifiability. The main result is also formalized and verified in Lean, following the approach described in Section 10.