  It is shown that the number of distinct types of three-point hinges, defined
by a real plane set of $n$ points is $\gg n^2\log^{-3} n$, where a hinge is
identified by fixing two pair-wise distances in a point triple. This is
achieved via strengthening (modulo a $\log n$ factor) of the Guth-Katz estimate
for the number of pair-wise intersections of lines in $\mathbb R^3$, arising in
the context of the plane Erd\H os distinct distance problem, to a second moment
incidence estimate. This relies, in particular, on the generalisation of the
Guth-Katz incidence bound by Solomon and Sharir.
