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D, E, I are collinear <=> P, Q, O are collinear

Source: Iran Third Round MO 1997, Exam 3, P5

June 30, 2012

geometrycircumcirclegeometry proposed

Problem Statement

In an acute triangle

ABC

let

AD

and

BE

be altitudes, and

AP

and

BQ

be bisectors. Let

I

and

O

be centers of incircle and circumcircle, respectively. Prove that the points

D, E

, and

I

are collinear if and only if the points

P, Q

, and

O

are collinear.