Please reference this problem.

In the above referenced problem a debate was raised as to the validity of given solution.  This debate caused me to ponder the following problem.

If you place 3 points in a line then obviously every line through any 2 points will intersect exactly 3 points.  Now the real question is, is it possible to place more than 3 points in a plane such that all lines between any 2 points intersect exactly 3 points, no more no less.

If the answer to the above question is in the negative, prove it. Otherwise, derive the appropriate example(s),

"H is an artificial point, where the perpendicular from A intersects BC. It may not be part of the N points."

I think the proof would be more complete by allowing H to coincide with one of the points. Then a case would be included where D coincides with H, and the same logic applies as for D between H and C or between H and B.

Without inclusion of this case, one could say "What if the shortest non-zero distance occurred where the perpendicular from the outside point coincided with one of the N points?"