  Let us consider Euclidean first-passage percolation on the Poisson-Delaunay
triangulation. We prove almost sure coalescence of any two semi-infinite
geodesics with the same asymptotic direction. The proof is based on an adapted
Burton-Keane argument and makes use of the concentration property for
shortest-path lengths in the considered graphs. Moreover, by considering the
specific example of the relative neighborhood graph, we illustrate that our
approach extends to further well-known graphs in computational geometry. As an
application, we show that the expected number of semi-infinite geodesics
starting at a given vertex and leaving a disk of a certain radius grows at most
sublinearly in the radius.
