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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 17, 1998 1:59 PM
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John Conway wrote,

> But as to a version for tetrahedra, I can't think of any reasonable
>type of solid-angle quadrisection that could even take part in a
>meaningful statement, let alone a true one! [Let me say that although
>lots of triangle-geometry does admit extensions to tetrahedra, there's
>lots that doesn't even among the very simple stuff - for instance the
>general tetrahedron doesn't have an orthocenter.]


Hmmm. Well, I was shooting from the hip, really all I'm capable of, being
largely ignorant of all these triangle centers, which seem to form such a
rich subject.

For a meaningful solid-angle quadrisection, all I can imagine at this
point, is, fixing one's attention upon a single vertex of a tetrahedron,
let the three faces surrounding be bisected at that vertex. Then the three
lines of bisection, with the three edges of the tetrahedron, define four
solid angles meeting at the original vertex: three on "the exterior," and a
single one interior.

Completing the construction, one obtains a dissection of the original
tetrahedron into 11 tetrahedra, one on each of the six original edges, four
hidden but touching each original vertex, and one central tetrahedron,
touching the centroid of each face. Four solid angles meet at each vertex.
What might one have to do, to make the four solid angles equal? Intercept
each tetrahedral region by a sphere on the common vertex, and give up on
"bisecting" the angles of the original tetrahedron, allowing the divisions
to fluctuate until the volumes of the tetrahedral regions meeting at a
vertex are equal, or ...?

--Russell Towle

Russell Towle
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Date Subject Author
9/12/98
Read A Theorem concerning the Trisectors of a Triangle
Den Roussel
9/12/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/13/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Larry Cusick
10/27/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/13/98
Read Re: A Theorem concerning the Trisectors of a Triangle
steve sigur
9/14/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/15/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/15/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Richard Guy
9/15/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/15/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Richard Guy
9/15/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Richard Guy
9/16/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Floor van Lamoen
9/16/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/17/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Floor van Lamoen
9/17/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/17/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Floor van Lamoen
9/17/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Russell Towle
9/17/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/17/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Russell Towle
9/17/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Douglas J. Zare
9/19/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Russell Towle
9/20/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/20/98
Read Re: A Theorem concerning the Trisectors of a Triangle
John Conway
9/18/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Antreas P. Hatzipolakis
11/10/98
Read Re: A Theorem concerning the Trisectors of a Triangle
Den Roussel

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