How many squares, solution

Time complexity: O(n^2 log n)

We first apply an angular sorting of the lines. Note that we may do it 
using brute-force, in quadratic time. Then we look at groups of parallel lines
in pairs: with one group A, we consider also the group B of the lines 
perpendicular to A. We generate all the distances between lines from A,
and sort them to figure out, for every distance d, how many pairs of lines
from A are in distance d from each other. Then we look at all the pairs of
lines from B, and for every pair that is in distance d we add to the result
the number of pairs of lines of A that have this distance (stored in a map
or binary searched, or whatever).

Note: one can change the problem to "How many rectangles", and then you can 
solve it in O(n log n) time.