DE-MO 2003 Problem 2

There are four circles $k_1 , k_2 , k_3$ and $k_4$ of equal radius inside the triangle $ABC$ . The circle $k_1$ touches the sides $AB, CA$ and the circle $k_4$ , $k_2$ touches the sides $AB,BC$ and $k_4$ , and $k_3$ touches the sides $AC, BC$ and $k_4.$ Prove that the center of $k_4$ lies on the line connecting the incenter and circumcenter of $ABC.$