IMO 1989 LL GRE29

Let L denote the set of all lattice points of the plane (points

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IMO 1989 LL GRE29

Origin: GRE

Problem

Let L denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points A, B, C of L there is a fourth point D, different from A, B, C, such that the interiors of the segments AD, BD, CD contain no points of L. Is the statement true if one considers four points of L instead of three?