  Planar point sets with many triple lines (which contain at least three
distinct points of the set) have been studied for 180 years, started with
Jackson and followed by Sylvester. Green and Tao has shown recently that the
maximum possible number of triple lines for an n element set is [n(n - 3)/6] +
1. Here we address the related problem of describing the structure of the
asymptotically near-optimal configurations, i.e., of those for which the number
of straight lines, which go through three or more points, has a quadratic
(i.e., best possible) order of magnitude. We pose the problem whether such
point sets must always be related to cubic curves. To support this conjecture
we settle various special cases; some of them are also related to the
four-in-a-line problem of Erd\H{o}s.
