IMO Shortlist 2004 problem G3


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2. travnja 2012.
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Let ABC be an acute-angled triangle such that \angle ABC<\angle ACB, let O be the circumcenter of triangle ABC, and let D=AO\cap BC. Denote by E and F the circumcenters of triangles ABD and ACD, respectively. Let G be a point on the extension of the segment AB beyound A such that AG=AC, and let H be a point on the extension of the segment AC beyound A such that AH=AB. Prove that the quadrilateral EFGH is a rectangle if and only if \angle ACB-\angle ABC=60^\circ.

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Official version Let O be the circumcenter of an acute-angled triangle ABC with {\angle B<\angle C}. The line AO meets the side BC at D. The circumcenters of the triangles ABD and ACD are E and F, respectively. Extend the sides BA and CA beyond A, and choose on the respective extensions points G and H such that {AG=AC} and {AH=AB}. Prove that the quadrilateral EFGH is a rectangle if and only if {\angle ACB-\angle ABC=60^{\circ }}.


Edited by orl.
Izvor: Međunarodna matematička olimpijada, shortlist 2004