  We consider the following iterative construction of a random planar
triangulation. Start with a triangle embedded in the plane. In each step,
choose a bounded face uniformly at random, add a vertex inside that face and
join it to the vertices of the face. After n-3 steps, we obtain a random
triangulated plane graph with n vertices, which is called a Random Apollonian
Network (RAN). We show that asymptotically almost surely (a.a.s.) every path in
a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also
show that a RAN always has a path of length (2n-5)^{log 2/log 3}, and that the
expected length of its longest path is Omega(n^0.88). Finally, we prove that
a.a.s. the diameter of a RAN is asymptotic to c log n, where c \approx 1.668 is
the solution of an explicit equation.
