  Let $X:=X_n\cup\{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson point
process of intensity $n$. We provide a first non-trivial lower bound for the
distance between the expected length of the shortest path between $(0,0)$ and
$(1,0)$ in the Delaunay triangulation associated with $X$ when the intensity of
$X_n$ goes to infinity. Experimental values indicate that the correct value is
about 1.04. We also prove that the expected number of Delaunay edges crossed by
the line segment $[(0,0),(1,0)]$ is equivalent to $2.16\sqrt{n}$ and that the
expected length of a particular path converges to 1.18 giving an upper bound on
the stretch factor.
