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Homothety with incenter and circumcenters

Source: Korea National Olympiad 2009 Problem 1

September 9, 2012
geometrygeometric transformationhomothetyincentercircumcircledilationratio

Problem Statement

Let I,OI, O be the incenter and the circumcenter of triangle ABCABC, and D,E,FD,E,F be the circumcenters of triangle BIC,CIA,AIB BIC, CIA, AIB. Let P,Q,R P, Q, R be the midpoints of segments DI,EI,FI DI, EI, FI . Prove that the circumcenter of triangle PQRPQR , MM, is the midpoint of segment IOIO.
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