Given a set P of n points in the plane and a set S of non-crossing line segments whose endpoints are in P, let CDT(P, S) be the constrained Delaunay triangulation of P with respect to S. Given any two visible points p, q \in P, we show that there exists a path from p to q in CDT(P, S), denoted SPCDT(p, q), such that every edge in the path has length at most |pq| and the ratio |SP CDT(p, q)|/|pq| is at most \frac{{4\pi \sqrt 3 }}{9}( \approx 2.42), thereby improving on the previously known bound of \frac{{\pi (1 + \sqrt 5 }}{2}( \approx 5.08).