2001 IMO Shortlist Problems/G1

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Problem

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.