DE-BUNDESWETTBEWERB 2004 Problem 7

Given two circles $k_1$ and $k_2$ which intersect at two different points $A$ and $B$ . The tangent to the circle $k_2$ at the point $A$ meets the circle $k_1$ again at the point $C_1$ . The tangent to the circle $k_1$ at the point $A$ meets the circle $k_2$ again at the point $C_2$ . Finally, let the line $C_1C_2$ meet the circle $k_1$ in a point $D$ different from $C_1$ and $B$ .

Prove that the line $BD$ bisects the chord $AC_2$ .