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hexagon area = 1/3 triangle area, plus concurrency question

Source: Mexican Mathematical Olympiad 1999 OMM P3

July 28, 2018
geometryareasconcurrency

Problem Statement

A point PP is given inside a triangle ABCABC. Let D,E,FD,E,F be the midpoints of AP,BP,CPAP,BP,CP, and let L,M,NL,M,N be the intersection points of BF BF and CE,AFCE, AF and CD,AECD, AE and BDBD, respectively. (a) Prove that the area of hexagon DNELFMDNELFM is equal to one third of the area of triangle ABCABC. (b) Prove that DL,EMDL,EM, and FNFN are concurrent.
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