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Concurrent lines

Source: Greek national M.O. 2009, Final Round,problem 2

November 15, 2011
geometrycircumcirclegeometric transformationhomothetygeometry unsolved

Problem Statement

Consider a triangle ABCABC with circumcenter OO and let A1,B1,C1A_1,B_1,C_1 be the midpoints of the sides BC,AC,AB,BC,AC,AB, respectively. Points A2,B2,C2A_2,B_2,C_2 are defined as OA2→=λ⋅OA1→, OB2→=λ⋅OB1→, OC2→=λ⋅OC1→,\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1}, where λ>0.\lambda >0. Prove that lines AA2,BB2,CC2AA_2,BB_2,CC_2 are concurrent.
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