In an acute-angled triangle let be altitudes and internal bisectors. Denote by and the incenter and the circumcentre of the triangle, respectively. Prove that the points and are collinear if and only if the points and are collinear.
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In an acute-angled triangle $ABC,$ let $AD,BE$ be altitudes and $AP,BQ$ internal bisectors. Denote by $I$ and $O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $D, E,$ and $I$ are collinear if and only if the points $P, Q,$ and $O$ are collinear.
Izvor: Međunarodna matematička olimpijada, shortlist 1997
Školjka 
let 
be altitudes and 
internal bisectors. Denote by 
and 
the incenter and the circumcentre of the triangle, respectively. Prove that the points 
and 
and