IMO Shortlist 1964 problem 5
Dodao/la:
arhiva2. travnja 2012. 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
?
%V0
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 
