RO-TST 2010 Problem 11

Let $ABC$ be a triangle such that $AB \neq AC$ . The internal bisector lines of the angles $ABC$ and $ACB$ meet the opposite sides of the triangle at points $B_0$ and $C_0$ , respectively, and the circumcircle $ABC$ at points $B_1$ and $C_1$ , respectively. Further, let $I$ be the incentre of the triangle $ABC$ . Prove that the lines $B_0C_0$ and $B_1C_1$ meet at some point lying on the parallel through $I$ to the line $BC$ .

Radu Gologan