IGO 2015 A4

In triangle $ABC$, we draw the circle with center $A$ and radius $AB$. This circle intersects $AC$ at two points. Also we draw the circle with center $A$ and radius $AC$ and this circle intersects $AB$ at two points. Denote these four points by $A_1, A_2, A_3, A_4$. Find the points $B_1, B_2, B_3, B_4$ and $C_1, C_2, C_3, C_4$ similarly. Suppose that these $12$ points lie on two circles. Prove that the triangle $ABC$ is isosceles.