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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)

Problem Statement