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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.

flag	asked Mar 14 2010 at 16:39 Mark B Villarino 109●3

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 2010 at 17:35

hello · Answer 1 · 2010-03-14 18:49:53Z UTC

Yes, it generalizes.

For any two nondegenerate tetrahedra A and B, you can find an affine transformation such that T(A)=B. Since affine transformations preserve ratios of line segments and areas and volumes, with a Routh's-theorem-type construction the ratio of volume of original tetrahedron to volume of inner polyhedron will be preserved.

As a commenter pointed out, with a tetrahedron there may be more than one way of defining a sensible cut. Once you've defined this, though, you can find a convenient tetrahedron (maybe a regular one, maybe one with a lot of right angles) and use it to calculate the proportion you want.

Routh’s theorem in three dimensions

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