
Re: A Theorem concerning the Trisectors of a Triangle
Posted:
Sep 19, 1998 12:38 PM


Douglas Zare, on September 17th 1998, wrote:
>If one trisects the dihedral angles as you suggest, do the opposite >endpoints of the rays of vertices of Morley triangles ever/always form the >vertices of an ideal regular icosahedron?
That is such a fascinating conjecture! I myself did not think of trisecting the dihedral angles of a tetrahedron; this may indeed may be the best 3D analogy to trisecting the interior angles of a triangle.
No, I did not think of this; instead, upon the analogy to a plane angle being trisected into three "regions of equal area" (supposing the regions were bounded by an arc of a circle centered upon the vertex), I strove to find a way to divide the solid angle at one vertex of a tetrahedron into "four regions of equal volume."
If one bisects each interior angle of the four triangles bounding a tetrahedron, then the bisectors meet in an interior point of each triangle (is this the "centroid"?). There are four of these points. These four centroids alone define the vertices of a smaller tetrahedron.
If we name the vertices of the initial tetrahedron V1, V2, V3, V4, and the four centroids of its faces C1,C2, C3, C4, then we can dissect it into eleven smaller tetrahedra as follows:
1. {C1, C2, C3, C4}.
2. {C1, C2, C3, V4}. 3. {C1, C2, C4, V3}. 4. {C1, C3, C4, V2}. 5. {C2, C3, C4, V1}.
6. {C1, C2, V3, V4}. 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.
Russell Towle Giant Gap Press: books on California history, digital topographic maps P.O. Box 141 Dutch Flat, California 95714  Voice: (916) 3892872 email: rustybel@foothill.net 

