HYACINTHOS 5246

Antreas P. Hatzipolakis

 

On Sunday, April 14, 2002, at 08:26 PM, Steve Sigur wrote:

> I have a serious reason for wanting this answer. In barycentric
> coordinates,
> most of the triangle centers are rational functions of the sides a, b,
> c. In
> the field extension sense, we also know the points that are made from
> adding
> root2 and root3 to the formulas (these are the Vecten-like and
> Fermat-like
> points). I was wondering if there were any root5 extensions that had any
> relevance?
>

I am not sure what you mean exactly, but
here is a first thought:

Construct regular pentagons on the sides of ABC.
Let A',B',C' be their "apexes".

The triangles ABC, A'B'C' are perspective
with barycentric perspector:

(1/(cotA +- cot72) ::) =

(1/(cotA +- sqrt(5 + 2sqrt5)) ::)

[+,- for outwardly,inwardly constructed 5gons)

APH