at least three *geometrically distinct* closed embedded geodesics. Unfortunately, the original proof by Lusternik-Schnirelmann [@Lusternik-Schnirelmann] contains a serious gap and various attempts have been made to fix it (see [@Taimanov92]). The fundamental issue there is *multiplicity*, that one of the geodesics obtained may just be a multiple cover of another geodesic. It is extremely technical (and many false proofs were given) to rule out this situation by modifying the method of Lusternik-Schnirelmann. In [@Grayson89], Grayson gave a rigorous proof of the Theorem of Three Geodesics by a careful analysis of the curve shortening flow on Riemannian surfaces. He proved that under the curve shortening flow, any embedded curve remains embedded and would either converge to a point in finite time or an embedded closed geodesic as time goes to infinity. As a curve which is initially embedded stays embedded throughout the flow, this prevents the multiplicity problem encountered by Lusternik-Schnirelmann’s approach using a discrete curve shortening process of Birkhoff [@Birkhoff17]. On the other hand, the situation in Theorem \[T:LS\] and \[T:double-normals\] are simpler as multiplicity cannot occur (see [@Giannoni-Majer97 Remark 3.2]).

We show that the convergence behavior for the chord shortening flow is similar to that for the curve shortening 