KöMaL 2012 A559

The incircle of triangle ABC is $k$. The circle $k_A$ touches $k$ and the segments $AB$ and $AC$ at at $A$', $A_B$ and $A_C$ respectively. The circles $k_B$, $k_C$ and the points $B$', $C$' are defined analogously. The second intersection point of the circles $A$'$B$'$A_B$ and $A$'$C$'$A_C$, other than $A$', is $K$. The line $A$'$K$ meets $k$ at $R$, other than $A$'. Prove that $R$ lies on the radical axis of the circles $k_B$ and $k_C$.