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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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John Conway

Posts: 2,238
Registered: 12/3/04
Re: A Theorem concerning the Trisectors of a Triangle
Posted: Sep 14, 1998 11:06 AM
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On Sun, 13 Sep 1998, Steve Sigur wrote:

> Den Roussel wrote
>

> > If the Angular trisectors of a triangle
> >are produced to the Circumcircle , then the Chords of adjacent
> >trisectors form an Equilateral triangle.

>
> i am unable to verify this, perhaps because i am not sure how to
> interpret "chords of adjacent bisectors.


I can help, since I've been playing with the theorem. In my figure:

B
/ \
A2 / \ C1
/ \
/ \
/ \
A1 / A0 C0 \ C2
/ \
/ B0 \
/ \
C-------------------A

B2 B1
(which I haven't been able to draw very accurately), A0 B0 C0
is the Morley triangle, and A B0 A1 and C B0 C2 are
"the adjacent trisectors" to the edge CA, with A1 and C2
being on the circumcircle. The edges of Den Roussel's triangle
are the three lines A1 C2, C1 B2, B1 A2.

It is indeed connected with the Morley triangle, corresponding
edges of the two triangles being parallel. But it doesn't seem
to be so "deep", since its equilaterality is much easier to prove.
I'm looking for some great generalization of the parallelness of
these two triangles.

John Conway




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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