Let project all segments to some line (x,y --> z=ax+by) and try all pairs of segments, which have an intersection of projections. To do it just sort events "open segment", "close segment" and brute force pairs.

You will get TL. Yet.

Now let try 4 lines: (a,b) = (1,0) (0,1) (1,1) (1,-1). At first, let estimate number of pairs to process for each line in O(nlogn) time. Now that we may choose line, which produces minimal mumber of pairs.

AC in 0.937 seconds. id = 4870936.