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Geometric Inequality on the lengths - ILL 1990 YUG1

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September 19, 2010
inequalitiesgeometrycircumcirclegeometry proposed

Problem Statement

Let S,TS, T be the circumcenter and centroid of triangle ABCABC, respectively. MM is a point in the plane of triangle ABCABC such that 90∘≤∠SMT<180∘90^\circ \leq \angle SMT < 180^\circ. A1,B1,C1A_1, B_1, C_1 are the intersections of AM,BM,CMAM, BM, CM with the circumcircle of triangle ABCABC respectively. Prove that MA1+MB1+MC1≥MA+MB+MC.MA_1 + MB_1 + MC_1 \geq MA + MB + MC.
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