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Geometry Difficulty 7.0 National Olympiad, round 2 Prove it Hungary

On the circumcircle of the scalene triangle $ABC$ , let $A_0$ , $B_0$ , and $C_0$ be the midpoints of the arcs $BAC$ , $CBA$ and $ACB$ , respectively. Denote by $A_1$ , $B_1$ and $C_1$ the Feuerbach points of the triangles $AB_0C_0$ , $BC_0A_0$ and $CA_0B_0$ , respectively. Show that the triangles $A_0B_0C_0$ and $A_1B_1C_1$ are similar.
Russian problem
(5 pont)

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