MathDB

Collinear iff the points P, Q, and O are collinear

Source: IMO Shortlist 1997, Q16

August 10, 2008

geometryincentercircumcircletrigonometryangle bisectorcollinearityIMO Shortlist

Problem Statement

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.