2004 OIM Problems/Problem 6

Problem

For a set $H$ of points in the plane, a point $P$ in the plane is said to be a "cut point" of $H$ if there are four different points $A, B, C$ and $D$ in $H$ such that the lines $AB$ and $CD$ are different and intersect at $P$ . Given a finite set $A_0$ of points in the plane, a sequence of sets is constructed $A_1, A_2, A_3, \cdots$ as follows: for any $j \ge 0$ , $A_{j+1}$ is the union of $A_j$ with the set of all cut points of $A_j$ . Show that if the union of all the sets of the sequence is a finite set, then for any $j \ge 1$ we have $A_j = A_1$ .