RO-NMO 2008 Problem 84

Let $ABCDA'B'C'D'$ be a cube. On the sides $(A'D')$ , $(A'B')$ and $(A'A)$ we consider the points $M_1$ , $N_1$ and $P_1$ respectively. On the sides $(CB)$ , $(CD)$ and $(CC')$ we consider the points $M_2$ , $N_2$ and $P_2$ respectively. Let $d_1$ be the distance between the lines $M_1N_1$ and $M_2N_2$ , $d_2$ be the distance between the lines $N_1P_1$ and $N_2P_2$ , and $d_3$ be the distance between the lines $P_1M_1$ and $P_2M_2$ . Suppose that the distances $d_1$ , $d_2$ and $d_3$ are pairwise distinct. Prove that the lines $M_1M_2$ , $N_1N_2$ and $P_1P_2$ are concurrent.