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Locus of a point satisfying a given relation

Source: Moldova IMO-BMO TST 2003, day 3, problem 3

August 16, 2008
geometrygeometry unsolved

Problem Statement

Consider a point M M found in the same plane with the triangle ABC ABC, but not found on any of the lines AB,BC AB,BC and CA CA. Denote by S1,S2 S_1,S_2 and S3 S_3 the areas of the triangles AMB,BMC AMB,BMC and CMA CMA, respectively. Find the locus of M M satisfying the relation: (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)
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