Trifold arrangements and cevian dissections

Abstract.

We determine precisely how many triple points can be formed in the plane by an arrangement of n lines lying in three parallel families of p, q, r lines, respectively. Using this result we solve the Euclidean realization problem for such arrangements. We apply our results to solve an analogous problem in which a triangle is dissected by three families of cevians. We conclude by mentioning some related unsolved problems.