Question 111
In the space, three segments $A_1A_2$, $B_1B_2$, and $C_1C_2$ are chosen.
They are not coplanar, and they meet at point $P$. Denote by $O_{ijk}$ the center of the sphere passing through $A_i, B_j, C_k$, and $P$. Prove that the four lines $O_{111}O_{222}, O_{112}O_{221}, O_{121}O_{212}$, and $O_{211}O_{122}$ have a common point.
They are not coplanar, and they meet at point $P$. Denote by $O_{ijk}$ the center of the sphere passing through $A_i, B_j, C_k$, and $P$. Prove that the four lines $O_{111}O_{222}, O_{112}O_{221}, O_{121}O_{212}$, and $O_{211}O_{122}$ have a common point.