# Counting points and lines in a plane

Let $P_1$ be a set of 4 points in the Euclidean plane. Formally, $P_1$ determines a set $L_1$ of 6 lines, which then determine only 3 points not already in $P_1.$ Let $P_2$ be the set of 7 points thus far determined. Formally, $P_2$ determines only 3 lines not already in $L_1.$ Let $L_2$ be the set of 9 lines thus far determined. Continuing in this manner determines sets $P_n$ and $L_n.$ What can be said about the cardinalities $|P_n|$ and $|L_n|$?

• I assume you want $P_1$ to be generic; the counts go down if three points are collinear or are vertices of a parallelogram (though the latter degeneracy can be removed by working in the projective plane instead of the Euclidean one: in the projective plane, all $P_1$ in general linear position are equivalent). – Noam D. Elkies Jul 18 '18 at 13:52

As Noam Elkies commented, some arrangements allow for three points on a line, or more than two lines meeting at a newly created point. Combinatorially, and assuming such coincidences do not occur, n points determine $m=\binom{n}{2}$ many lines with each of the $n$ points on $n-1$ distinct lines, and the $m$ lines then determine $m(m -2n+3)/2$ new points, as a line intersects $2n-4$ other lines in its two old points.