  A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every
line is incident with $k$ points, every point is incident with $r$ lines, and
for each point $x$, there is a line incident with precisely those points that
are not collinear with $x$. Here we generalize the concept by allowing the
points not collinear with $x$ to form the point set of a Steiner system
$S(2,k,w)$ whose blocks are lines of the geometry.
