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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 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

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