  We present a deterministic local routing algorithm that is guaranteed to find
a path between any pair of vertices in a half-$\theta_6$-graph (the
half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the
empty region is an equilateral triangle). The length of the path is at most
$5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of
vertices. Moreover, we show that no local routing algorithm can achieve a
better routing ratio, thereby proving that our routing algorithm is optimal.
This is somewhat surprising because the spanning ratio of the
half-$\theta_6$-graph is 2, meaning that even though there always exists a path
whose lengths is at most twice the Euclidean distance, we cannot always find
such a path when routing locally.
  Since every triangulation can be embedded in the plane as a
half-$\theta_6$-graph using $O(\log n)$ bits per vertex coordinate via
Schnyder's embedding scheme (SODA 1990), our result provides a competitive
local routing algorithm for every such embedded triangulation. Finally, we show
how our routing algorithm can be adapted to provide a routing ratio of
$15/\sqrt{3} \approx 8.660$ on two bounded degree subgraphs of the
half-$\theta_6$-graph.
