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China Mathematical Olympiad 1991 problem1

Source: China Mathematical Olympiad 1991 problem1

October 8, 2013
geometry

Problem Statement

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