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Topic: A Theorem concerning the Trisectors of a Triangle
Replies: 24   Last Post: Nov 10, 1998 12:19 AM

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 Russell Towle Posts: 70 Registered: 12/3/04
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
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Date Subject Author
9/12/98 Den Roussel
9/12/98 John Conway
9/13/98 Larry Cusick
10/27/98 John Conway
9/13/98 steve sigur
9/14/98 John Conway
9/15/98 John Conway
9/15/98 Richard Guy
9/15/98 John Conway
9/15/98 Richard Guy
9/15/98 Richard Guy
9/16/98 Floor van Lamoen
9/16/98 John Conway
9/17/98 Floor van Lamoen
9/17/98 John Conway
9/17/98 Floor van Lamoen
9/17/98 Russell Towle
9/17/98 John Conway
9/17/98 Russell Towle
9/17/98 Douglas J. Zare
9/19/98 Russell Towle
9/20/98 John Conway
9/20/98 John Conway
9/18/98 Antreas P. Hatzipolakis
11/10/98 Den Roussel