Shamos-Hoey Line segment intersection runtime
The BST is ordered with a "above/below" relationship. The idea is that for any "event", the BST is ordered by the y-value where each line segment intersects the x-value of that "event". 

When a segment is inserted or deleted, the comparisons in the BST are between the y-value of each line segment at the x-value of the current event. This can be computed with geometry in constant time given the slope of each line. The ordering only requires a single update at each event (a little subtle); the BST requires no other maintenance.

In your example, at the 2nd event when C is added, it is actually inserted above A. Then the B/C intersection would be found correctly.