ILL 1989 Problem 27

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?