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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 15, 1998 10:06 AM
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On 12 Sep 1998, Den Roussel wrote (and I've edited for greater clarity):

> If the Angular trisectors of a triangle are produced to the Circumcircle,
> then the Chords determined by the pairs of trisectors adjacent to the
> edges form an Equilateral triangle.
>
> It has been suggested that this result might follow from Morleys
> theorem. So far, I have been unable to make this connection. Any
> thoughts ?


As I've already remarked, this is indeed connected to Morley's theorem,
in that Den Roussel's triangle is parallel to Morley's. However, I can
produce no deduction of this result from Morley's theorem that's any
simpler than an outright proof of it not using that theorem.

I conjectured that the parallelism might continue to hold if the
trisectors of the typical angle (say B) were replaced by arbitrary
pairs of isogonal lines (ie., B B1 and B B2 such that angles A B B1
and C B B2 are equal). The main purpose of this note is to announce
that I've now disproved this conjecture (by replacing the Morley triangle
by the Brocard "minor triangle").

However, I'm still very hopeful that there will exist SOME
wonderful generalization, because this construction is near others
that have interesting properties. For instance, it is known that
if A(P), B(P), C(P) are the points where the Cevians of P hit
the circumcircle again, then the triangle they form is similar to
the pedal triangle of P (whose vertices A[P], B[P], C[P] are
where the normals from P hit the sides).

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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