Question: Problem in combinatorial geometry

Let four straight lines in general position be given. Their complement consists of  11 domains. One can choose four (PS. different) points belonging to the pieces s.t. exactly two points lie astride everyone of these lines. Here is an example.

 

restart; with(geometry)``

line(L1, [point(A, 1, 1/2), point(B, 2, 1)]):

geometry:-line(L2, [geometry:-point(C, 1, -1/2), geometry:-point(E, 2, -1)]):

geometry:-line(L3, [geometry:-point(F, 1/2+1, 1), geometry:-point(G, 1+1, 2)]):

geometry:-line(L4, [geometry:-point(H, -1/2+2, 1+2), geometry:-point(K, (-1)+2, 2+2)]):

geometry:-point(P1, [-6, 0]):``

geometry:-point(P2, [2, 6]):

geometry:-point(P3, [8, -6]):

geometry:-draw([L1, L2, L3, L4, P1(color = blue, symbol = solidcircle, symbolsize = 15), P2(color = blue, symbol = solidcircle, symbolsize = 15), P3(color = blue, symbol = solidcircle, symbolsize = 15), P4(color = blue, symbol = solidcircle, symbolsize = 15)])

 

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Download four_points.mw




How many are such configurations (If the points belong to the same pieces, two configurations are equivalent. PS. I.e. one may move points in the  pieces.)?
How to determine it with Maple? Robert Israel used GraphTheory, having answered that.

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