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ASU 368 All Soviet Union MO 1983 maximal sidelength inequality

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July 28, 2019
geometric inequalityinequalitiesgeometry

Problem Statement

The points D,E,FD,E,F belong to the sides (AB),(BC)(AB), (BC) and (CA)(CA) of the triangle ABCABC respectively (but they are not vertices). Let us denote with d0,d1,d2d_0, d_1, d_2, and d3d_3 the maximal side length of the triangles DEFDEF, DEADEA, DBFDBF, CEFCEF, respectively. Prove that d0≥32min{d1,d2,d3}d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\} When the equality takes place?
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