Any intersecting segments

Sort segments by left endpoints

Pass a vertical sweep line over the segments

Put segment in RB tree when hit left endpoint, order by y value
Check intersection between new and ABOVE and between new and BELOW

Remove segment from RB tree when hit right endpoint
Check intersection between ABOVE and BELOW

If two line segments intersect, they will eventually be neighbors

Neighbors when one of the segments is added, and the first check will catch the intersection

Neighbors when another segment is deleted, and the second check will catch the intersection

The intersecting lines will eventually intersect the sweep line.

No false positives

Assume no vertical lines, no 3 lines intersect at same point

Sweep line: RB tree contains 1
Sweep line: RB tree contains 2, 1, no intersect
Sweep line: RB tree contains 2, 1, 3, no intersect
Sweep line: RB tree contains 2, 3, intersect
Sweep line: RB tree contains 2, 4, 3, no intersect
Sweep line: RB tree contains 2, 4, 3, 5, no intersect
Sweep line: RB tree contains 2, 4, 3, no intersect
Sweep line: RB tree contains 2, 3, intersect
Sweep line: RB tree is empty

   For n segments:
      sort segments                n lg n
      n RB-Inserts                 n lg n
      n RB-Deletes                 n lg n
      O(1) comparison

The algorithm takes O(n lg n) time.

Scanline Algorithm Applet