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The cube

Source: RMO 2008, Grade 8, Problem 4

April 30, 2008
geometry3D geometry

Problem Statement

Let ABCDA′B′C′D′ ABCDA'B'C'D' be a cube. On the sides (A′D′) (A'D'), (A′B′) (A'B') and (A′A) (A'A) we consider the points M1 M_1, N1 N_1 and P1 P_1 respectively. On the sides (CB) (CB), (CD) (CD) and (CC′) (CC') we consider the points M2 M_2, N2 N_2 and P2 P_2 respectively. Let d1 d_1 be the distance between the lines M1N1 M_1N_1 and M2N2 M_2N_2, d2 d_2 be the distance between the lines N1P1 N_1P_1 and N2P2 N_2P_2, and d3 d_3 be the distance between the lines P1M1 P_1M_1 and P2M2 P_2M_2. Suppose that the distances d1 d_1, d2 d_2 and d3 d_3 are pairwise distinct. Prove that the lines M1M2 M_1M_2, N1N2 N_1N_2 and P1P2 P_1P_2 are concurrent.
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