Question 6
(POL) Given a tetrahedron $ABCD$, let $D_1$ be the centroid of the triangle $ABC$ and let $A_1,B_1,C_1$ be the intersection points of the lines parallel to $DD_1$ and passing through the points $A,B,C$ with the opposite faces of the tetrahedron. Prove that the volume of the tetrahedron $ABCD$ is one-third the volume of the tetrahedron $A_1B_1C_1D_1$. Does the result remain true if the point $D_1$ is replaced with any point inside the triangle $ABC$?