AUSTRIAN-POLISH 2003 Problem 6

$ABCD$ is a tetrahedron such that we can find a sphere $k(A,B,C)$ through $A, B, C$ which meets the plane $BCD$ in the circle diameter $BC$ , meets the plane $ACD$ in the circle diameter $AC$ , and meets the plane $ABD$ in the circle diameter $AB$ . Show that there exist spheres $k(A,B,D)$ , $k(B,C,D)$ and $k(C,A,D)$ with analogous properties.