IMO Shortlist 1964 problem 5

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In tetrahedron ABCD, vertex D is connected with D_0, the centrod if \triangle ABC. Line parallel to DD_0 are drawn through A,B and C. These lines intersect the planes BCD, CAD and ABD in points A_2, B_1, and C_1, respectively. Prove that the volume of ABCD is one third the volume of A_1B_1C_1D_0. Is the result if point D_o is selected anywhere within \triangle ABC?
Source: Međunarodna matematička olimpijada, shortlist 1964