IZhO 2009 Problem 4

On the plane, a Cartesian coordinate system is chosen. Given points $A_1,A_2,A_3,A_4$ on the parabola $y = x^2$ , and points $B_1,B_2,B_3,B_4$ on the parabola $y = 2009x^2$ . Points $A_1,A_2,A_3,A_4$ are concyclic, and points $A_i$ and $B_i$ have equal abscissas for each $i = 1,2,3,4$ . Prove that points $B_1,B_2,B_3,B_4$ are also concyclic.