## points of concurrency and sets of parallel lines

My question is mainly concerned with discovering the allowable set of "configurations" of the given problem:

We have a two-dimensional board composed of three sets (of infinite size) of parallel lines \P_1, \P_2, \P_3, where the lines in \P_2 form a 60 degree angle with lines in \P_3 and \P_1, and lines in \P_1 and \P_3 are 120 degrees apart. Furthermore, any point of intersection of any two sets \P_i, \P_j has a unique line from \P_k intersecting it. Thus, our set of points of intersection are each composed exactly of three lines: one from each set.

For a visualization of the board (choose any empty hexagon cell and the three lines on which it lies are each from the different sets I described. Each empty hexagon cell is a point of intersection):

My question is the following: which are the impossible configurations of this board, in terms of specifying a set of intersections? To illustrate the question, consider the following example:

We will refer to a point of intersection as (i,j,k) where i is a line in P_1, j is a line in P_2 and k in P_3. Note that (i,j,k) is unique for every point.

I give the following set of points:

(i, i, i), (j, j, j), (k, k, k), (i, j, k), (j, k, i), (k, i, j), (i, k, j). This works out fine. However, when I add the following point (k, j, i) as a constraint, it appears impossible to interpret it on that board. This appears to be the case no matter what i, j, k are.

Do you have any idea what are the pathological cases (such as in the example), is there any theorem, paper that discusses the problem I have described?

Thank you.
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