Routh's theorem in three dimensions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:00:38Z http://mathoverflow.net/feeds/question/18179 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18179/rouths-theorem-in-three-dimensions Routh's theorem in three dimensions Mark B Villarino 2010-03-14T16:39:38Z 2010-03-14T18:49:53Z <p>The most well known case of Routh's triangle theorem is: <em>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.</em></p> <p>Here is my question: <strong><em>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?</em></strong></p> <p>As far as I know, this question has never been contemplated in the literature.</p> http://mathoverflow.net/questions/18179/rouths-theorem-in-three-dimensions/18187#18187 Answer by hello for Routh's theorem in three dimensions hello 2010-03-14T17:56:08Z 2010-03-14T18:49:53Z <p>Yes, it generalizes. </p> <p>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. </p> <p>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.</p>