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IO Euler line of triangle PQR

Source: Bosnia and Herzegovina TST 2013 problem 6

May 20, 2013
Eulergeometrycircumcirclegeometry proposed

Problem Statement

In triangle ABCABC, II is the incenter. We have chosen points P,Q,RP,Q,R on segments IA,IB,ICIA,IB,IC respectively such that IP⋅IA=IQ⋅IB=IR⋅ICIP\cdot IA=IQ \cdot IB=IR\cdot IC. Prove that the points II and OO belong to Euler line of triangle PQRPQR where OO is circumcenter of ABCABC.
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