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D,E,F on sides BC, CA, AB so that (AFE)=(BFD)=(CDE), prove (DEF)/(ABC)>=1/4

Source: Rioplatense Olympiad 2000 level 3 P2

September 4, 2018
geometrygeometric inequalityareas

Problem Statement

In a triangle ABCABC, points D,ED, E and FF are considered on the sides BC,CABC, CA and ABAB respectively, such that the areas of the triangles AFE,BFDAFE, BFD and CDECDE are equal. Prove that (DEF)(ABC)≥14\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}
Note: (XYZ)(XYZ) is the area of triangle XYZXYZ.
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