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FIGO is perpendicular

Source: 2018 Taiwan TST Round 3

April 2, 2020
geometryincentercircumcircleTaiwan

Problem Statement

Let I,G,OI,G,O be the incenter, centroid and the circumcenter of triangle ABCABC, respectively. Let X,Y,ZX,Y,Z be on the rays BC,CA,ABBC, CA, AB respectively so that BX=CY=AZBX=CY=AZ. Let FF be the centroid of XYZXYZ.
Show that FGFG is perpendicular to IOIO.
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