
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:
> I have not yet read any notification of the fact that in stead of inside > 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 angletriples does it happen?
John Conway

