We study online routing algorithms on the $Θ 6$-graph and the half-$Θ 6$-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex s and a target vertex t in the $Θ 6$-graph (resp. half-Θ 6-graph), there exists a deterministic online routing algorithm that finds a path from s to t whose length is at most 2 st (resp. 2.89 st) which is optimal in the worst case [Bose et al., siam J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the $Θ 6$-graph and half-$Θ 6$-graph defined on a Poisson point process.