  Assume we are given a set of parallel line segments in the plane, and we wish
to place a point on each line segment such that the resulting point set
maximizes or minimizes the area of the largest or smallest triangle in the set.
We analyze the complexity of the four resulting computational problems, and we
show that three of them admit polynomial-time algorithms, while the fourth is
NP-hard. Specifically, we show that maximizing the largest triangle can be done
in $O(n^2)$ time (or in $O(n \log n)$ time for unit segments); minimizing the
largest triangle can be done in $O(n^2 \log n)$ time; maximizing the smallest
triangle is NP-hard; but minimizing the smallest triangle can be done in
$O(n^2)$ time. We also discuss to what extent our results can be generalized to
polygons with $k>3$ sides.
