
Re: A Theorem concerning the Trisectors of a Triangle
Posted:
Sep 20, 1998 12:50 PM


> eleven smaller tetrahedra as follows: > 1. {C1, C2, C3, C4}.> 2. {C1, C2, C3, V4}.> 3. {C1, C2, C4, V3}. > 7. {C2, C3, V1, V4}.> 8. {C3, C4, V1, V2}.> 9. {C1, C3, V2, V4}. > 10. {C1, C4, V2, V3}.> 11. {C2, C4, V1, V3}.
> The question is, whether in general it is even possible for the four > tetrahedra which share one of the vertices of {V1, V2, V3, V4}, to be of > equal volume. For instance, numbers 5, 7, 8, and 11 of the list above share > vertex V1. Perhaps it is true that the *regions they represent* do have > equal volumes; for if one takes a section of the large tetrahedron, {V1, > V2, V3, V4},by a plane containing the triangle {C2, C3, C4}, and allows it > to cut off the four tetrahedra meeting at V1, then, I think, the four > triangles in the section are similar, and have equal area, and the height > to V1 is the same for all of them. Maybe.
Sorry my previous message was incomplete. This configuration looks interesting  I'll study it.
John Conway

