  Stationary Poisson processes of lines in the plane are studied whose
directional distributions are concentrated on $k \ge 3$ equally spread
directions. The random lines of such processes decompose the plane into a
collection of random polygons, which form a so-called Poisson line
tessellation. The focus of this paper is to determine the proportion of
triangles in such tessellations, or equivalently, the probability that the
typical cell is a triangle. As a by-product, a new deviation of Miles'
classical result for the isotropic case is obtained by an approximation
argument.
