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IMO ShortList 2001, geometry problem 3

Source: IMO ShortList 2001, geometry problem 3

September 30, 2004
geometryparallelogramminimizationTriangleIMO Shortlist

Problem Statement

Let ABCABC be a triangle with centroid GG. Determine, with proof, the position of the point PP in the plane of ABCABC such that AP⋅AG+BP⋅BG+CP⋅CGAP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG is a minimum, and express this minimum value in terms of the side lengths of ABCABC.
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