MathDB

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

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