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Centroid is a fixed point

Source: 2012 China TST - Quiz 1 - Day 2 - P4

March 15, 2012
geometrycircumcirclegeometry proposedInversion

Problem Statement

Given two circles ω1,ω2{\omega _1},{\omega _2}, SS denotes all ΔABC\Delta ABC satisfies that ω1{\omega _1} is the circumcircle of ΔABC\Delta ABC, ω2{\omega _2} is the AA- excircle of ΔABC\Delta ABC , ω2{\omega _2} touches BC,CA,ABBC,CA,AB at D,E,FD,E,F. SS is not empty, prove that the centroid of ΔDEF\Delta DEF is a fixed point.
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