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PAB and PCD have equal areas in convex ABCD

Source: Netherlands - Dutch NMO 1967 p1

January 31, 2023
geometryequal areasarea of a triangle

Problem Statement

In this exercise we only consider convex quadrilaterals.
(a) For such a quadrilateral ABCDABCD, determine the set of points PP contained within that quadrilateral for which PAPA and PCPC divide the quadrilateral into two pieces of equal areas.
(b) Prove that there is a point PP inside such a quadrilateral, such that the triangles PABPAB and PCDPCD have equal areas, as well as the triangles PBCPBC and PADPAD.
(c) Find out which quadrilaterals ABCDABCD contains a point PP, so that the triangles PABPAB, PBCPBC, PCDPCD and PDAPDA have equal areas.
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