ÖMO 1999 Problem 2

Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$ . The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$ , respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$ . Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$ . Describe the locus of the intersection point.