IMO 1964 Problem 6

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 $\triangle ABC$ ?