OIM 2004 Problem 6

Given a set $\mathcal{H}$ of points in the plane, $P$ is called an "intersection point of $\mathcal{H}$ " if distinct points $A,B,C,D$ exist in $\mathcal{H}$ such that lines $AB$ and $CD$ are distinct and intersect in $P$ . Given a finite set $\mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $j\geq0$ , $\mathcal{A}_{j+1}$ is the union of $\mathcal{A}_{j}$ and the intersection points of $\mathcal{A}_{j}$ . Prove that, if the union of all the sets in the sequence is finite, then $\mathcal{A}_{i}=\mathcal{A}_{1}$ for any $i\geq1$ .