  We consider the following problem: Let $\mathcal{L}$ be an arrangement of $n$
lines in $\mathbb{R}^3$ colored red, green, and blue. Does there exist a
vertical plane $P$ such that a line on $P$ simultaneously bisects all three
classes of points in the cross-section $\mathcal{L} \cap P$? Recently, Schnider
[SoCG 2019] used topological methods to prove that such a cross-section always
exists. In this work, we give an alternative proof of this fact, using only
methods from discrete geometry. With this combinatorial proof at hand, we
devise an $O(n^2\log^2(n))$ time algorithm to find such a plane and the
bisector of the induced cross-section. We do this by providing a general
framework, from which we expect that it can be applied to solve similar
problems on cross-sections and kinetic points.
