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areas of triangles

Source: Mathematics Regional Olympiad of Mexico Southeast 2016 P4

October 26, 2021
geometryalgebraquadrilateral

Problem Statement

The diagonals of a convex quadrilateral ABCDABCD intersect in EE. Let S1,S2,S3S_1, S_2, S_3 and S4S_4 the areas of the triangles AEB,BEC,CED,DEAAEB, BEC, CED, DEA respectively. Prove that, if exists real numbers w,x,yw, x, y and zz such that
S1=x+y+xy,S2=y+z+yz,S3=w+z+wz,S4=w+x+wx,S_1=x+y+xy, S_2=y+z+yz, S_3=w+z+wz, S_4=w+x+wx,
then EE is the midpoint of ACAC or EE is the midpoint of BDBD.
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