The Bentley–Ottmann algorithm finds all the intersections that occur in a set of line segments. Drag the endpoints of the line to move, click to add and right click to remove.
Instead of calculating the intersection of every pair of segments, which would require O(n²) comparisons for a set of n lines, the algorithm uses a sweep line. Starting at the top of the screen and moving down, it keeps track of the order of the lines' x positions at the current y position (represented by the figure on the right). As lines start, end or intersect, new pairs of lines become adjacent in the x ordering and are checked for intersections.
A set of lines with I intersections ends up only requiring O(n log n + I log n) operations. Computational Geometry: Algorithms and Applications, chapter 2 has a proof and a more detailed explanation.