RU-ALLRUSSIAN 2016 Problem 112

In the space given three segments $A_1A_2, B_1B_2$ and $C_1C_2$ , do not lie in one plane and intersect at a point $P$ . Let $O_{ijk}$ be center of sphere that passes through the points $A_i, B_j, C_k$ and $P$ . Prove that $O_{111}O_{222}, O_{112}O_{221}, O_{121}O_{212}$ and $O_{211}O_{122}$ intersect at one point. (P.Kozhevnikov)