IMO Shortlist 2001 problem G1

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Dodao/la: arhiva
April 2, 2012
Let A_1 be the center of the square inscribed in acute triangle ABC with two vertices of the square on side BC. Thus one of the two remaining vertices of the square is on side AB and the other is on AC. Points B_1,\ C_1 are defined in a similar way for inscribed squares with two vertices on sides AC and AB, respectively. Prove that lines AA_1,\ BB_1,\ CC_1 are concurrent.
Source: Međunarodna matematička olimpijada, shortlist 2001