CMO 1991 Problem 1

We are given a convex quadrilateral $ABCD$ in the plane. (i) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$ ? (ii) Find (with proof) the maximum possible number of such point $P$ which satisfies the condition in (i).