The number of lines determined by a set of points

[GraemeMcRae]

A set of n points can be said to "determine" a set of m lines such that each pair of points is contained in exactly one line, and each line passes through at least two points. If n=5, then m is an element of {1, 5, 6, 8, 10}.

Similarly, a set of 6 points determines m lines, where m can be any of {1, 6, 7, 8, 9, 10, 11, 13, 15}

A description of each of these configurations follows:

1: All 6 points on a single line
6: 5 points on a line, and 1 point not on the line
7: 3 points in each of four lines. Every point is contained in exactly two of the four 3-point lines and exactly one other (2-point) line.
8: 4 points on one line, 3 points on another, sharing one point
9: 3 points in each of three lines. Each pair of lines shares a point.
10: 4 points on a line, and two points off the line, neither of which is contained in a multi-point line
11: 3 points on one line, 3 points on another, the two lines sharing a common point, and one other point not contained in any multi-point line
11: 3 points on one line, 3 points on another, the two lines not sharing a common point
13: 3 points on a line, and three points off the line, none of which is contained in a multi-point line
15: No multi-point line

It's interesting to note that there are two distinct configurations of points and lines ("geometries") that have 11 lines.

OEIS A001200 tells you the number of geometries for each number of points, but it doesn't give a clue how the sequence is generated, nor how many lines are in each of the geometries, nor whether the configuration is possible in Euclidean space.

I'm not sure how this would become a puzzle, but Euler may think of a question that requires a programatical understanding of these geometries.