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5

Part of 2008 IberoAmerican

Problems(1)

Iberoamerican Olympiad 2008, problem 5

Source:

9/24/2008
Let ABC ABC a triangle and X X, Y Y and Z Z points at the segments BC BC, AC AC and AB AB, respectively.Let A′ A', B′ B' and C′ C' the circuncenters of triangles AZY AZY,BXZ BXZ,CYX CYX, respectively.Prove that 4(A′B′C′)≥(ABC) 4(A'B'C')\geq(ABC) with equality if and only if AA′ AA', BB′ BB' and CC′ CC' are concurrents. Note: (XYZ) (XYZ) denotes the area of XYZ XYZ
geometrygeometric transformationhomothetyratiocircumcircleanalytic geometryinequalities