IR-MO 2003 Problem 20

Suppose that $M$ is an arbitrary point on side $BC$ of triangle $ABC$ . $B_1,C_1$ are points on $AB,AC$ such that $MB = MB_1$ and $MC = MC_1$ . Suppose that $H,I$ are orthocenter of triangle $ABC$ and incenter of triangle $MB_1C_1$ . Prove that $A,B_1,H,I,C_1$ lie on a circle.