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Centroid and area

Source: Gulf MO 2012, Problem 1

June 14, 2012
geometryanalytic geometrygeometry proposed

Problem Statement

Let X, YX,\ Y and ZZ be the midpoints of sides BC, CABC,\ CA, and ABAB of the triangle ABCABC, respectively. Let PP be a point inside the triangle. Prove that the quadrilaterals AZPY, BXPZAZPY,\ BXPZ, and CYPXCYPX have equal areas if, and only if, PP is the centroid of ABCABC.
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