In tetrahedron , vertex is connected with , the centrod if . Line parallel to are drawn through and . These lines intersect the planes and in points and , respectively. Prove that the volume of is one third the volume of . Is the result if point is selected anywhere within ?
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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 $\triangle ABC$?
Izvor: Međunarodna matematička olimpijada, shortlist 1964
Školjka 
, vertex 
is connected with 
, the centrod if 
. Line parallel to 
are drawn through 
and 
. These lines intersect the planes 
and 
in points 
and 
, respectively. Prove that the volume of 
. Is the result if point 
is selected anywhere within