Routh's theorem in three dimensions

The most well known case of Routh's triangle theorem is:

If the sides BC, CA,and AB are trisected at the points D, E, and F, respectively, then the area of the inside triangle formed by AD, BE, CF is $\dfrac{1}{7}$th of the area of that of the triangle ABC.

Here is my question: can Routh's theorem be generalized to a tetrahedron which is cut by 4 planes through its 4 vertices and cutting the opposite faces appropriately?

As far as I know, this question has never been contemplated in the literature.

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You should tell us how you want the 4 planes to cut the opposite faces. If the tetrahedron has vertices ABCD, and you want to take a plane through A which is parallel to one of the edges of the face BCD, then you need to choose that edge. Note that this will leave at least two edge unchosen however you do it. –  Andrew Lobb Mar 14 '10 at 17:35