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An acute triangle ABC is given. Points A_1 and A_2 are taken on the side BC (with A_2 between A_1 and C), B_1 and B_2 on the side AC (with B_2 between B_1 and A), and C_1 and C_2 on the side AB (with C_2 between C_1 and B) so that

\angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1.

The lines AA_1,BB_1, and CC_1 bound a triangle, and the lines AA_2,BB_2, and CC_2 bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

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