Τρίτη, 17 Οκτωβρίου 2017

2006 IMO Shortlist G9

Points $A_1,B_1, C_1$ are chosen on the sides $BC, CA, AB$ of a triangle $ABC$, respectively. The circumcircles of triangles $AB_1C_1, BC_1A_1, CA_1B_1$ intersect the circumcircle of triangle ABC again at points $A_2, B_2, C_2$, respectively ($A_2 \ne A, B_2 \ne B, C_2 \ne C$). Points $A_3, B_3, C_3$  are symmetric to $A_1, B_1, C_1$ with respect to the midpoints of the sides $BC, CA, AB$ respectively. Prove that the triangles $A_2B_2C_2$ and $A_3B_3C_3$ are similar.

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