Since the notion of curved path (maybe aside from circular arcs) is not discussed in basic geometry, the problem would be impossible to state. If you assume gi

Since the notion of curved path (maybe aside from circular arcs) is not discussed in basic geometry, the problem would be impossible to state. If you assume given the notion of arc-length as the least-upper-bound on lengths of piecewise-linear paths between the two points, then the triangle inequality of basic geometry suffices. This seems not more hand-wavy a proof as something about "light-travelling-at-constant-speed" analogies.