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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 16, 1998 7:15 PM

On Wed, 16 Sep 1998, Floor van Lamoen wrote:

> trisectors also outside trisectors can be used. The equilateral triangle
> you get from the outside trisectors and the one you get from the inside
> trisectors are reflections of each other over the circumcenter.

I take it that this refers to Roussel's theorem rather than Morley's?
Actually, I can almost see the proof, so hardly needed to ask.

An interesting question is how many other equilateral triangle
theorems like Morley's and Napoleon's there are? Erect triangles
on the edges of ABC with base angles alpha at A, beta at B, gamma
at C. Then the apices of these form an equilateral triangle if the triple
alpha,beta,gamma is pi/6,pi/6,pi/6 (Napoleon) or -A/3,-B/3,-C/3
(Morley). This also happens for -pi/6,-pi/6,-pi/6 and for various
triples -A'/3,-B'/3,-C'/3 with A',B',C' congruent to A,B,C modulo pi.

For what other angle-triples does it happen?

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