Bisecting three classes of lines

We consider the following problem: Let L be an arrangement of n lines in R 3 general position colored red, green, and blue. Does there exist a vertical plane P such that line simultaneously bisects all three classes points induced by intersection with ? Recently, Schnider used topological methods to prove cross-section always exists. In this work, we give alternative proof fact, using only from discrete geometry. With combinatorial at hand, devise O ( 2 log ? ) time algorithm find bisector cross-section. do providing framework, which expect it can applied solve similar problems on cross-sections kinetic points.