  We present subquadratic algorithms in the algebraic decision-tree model for
several \textsc{3Sum}-hard geometric problems, all of which can be reduced to
the following question: Given two sets $A$, $B$, each consisting of $n$
pairwise disjoint segments in the plane, and a set $C$ of $n$ triangles in the
plane, we want to count, for each triangle $\Delta\in C$, the number of
intersection points between the segments of $A$ and those of $B$ that lie in
$\Delta$. The problems considered in this paper have been studied by
Chan~(2020), who gave algorithms that solve them, in the standard real-RAM
model, in $O((n^2/\log^2n)\log^{O(1)}\log n)$ time. We present solutions in the
algebraic decision-tree model whose cost is $O(n^{60/31+\varepsilon})$, for any
$\varepsilon>0$.
  Our approach is based on a primal-dual range searching mechanism, which
exploits the multi-level polynomial partitioning machinery recently developed
by Agarwal, Aronov, Ezra, and Zahl~(2020).
  A key step in the procedure is a variant of point location in arrangements,
say of lines in the plane, which is based solely on the \emph{order type} of
the lines, a "handicap" that turns out to be beneficial for speeding up our
algorithm.
