Date: Tue, 21 Oct 1997 07:51:37 -0600
From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: Triangle problem
Newsgroups: sci.math

In article <62grbc$rod@eng-ser1.erg.cuhk.edu.hk>,
  Peter Tam <tstam@cse.cuhk.edu.hk> wrote:
> Hi all,
>
>   Here is a problem that I don't know whether
> there is a solution or not, but it seems to have:
>
>   A triangle ABC.  Each angle < 120 degree.
>   The three vertices are: (Xa, Ya), (Xb, Yb), (Xc, Yc)
>
>   Now I want to find a point within a triangle,
> let it be (Xo, Yo).  This point satisfy the following:
> We draw three lines:  (Xo, Yo) - (Xa, Ya)
> and (Xo, Yo) - (Xb, Yb)
> and (Xo, Yo) - (Xc, Yc).
> The three lines form something like:
>
>   y | x
>    / \
>     z
>
> And three angles on this shape, say x, y, z, are
> all equal to 120 degree.
>
> So, is there any method to find (Xo, Yo) that
> satisfy the above ?
>
> (a recommanded book is also ok)

This is the Fermat point of the triangle. To construct it erect
equilateral triangles outward with the sides of the original triangle as
bases. Then join each vertex of the original triangle to the far vertex
of the equilateral triangle built on the opposite side. One gets three
lines this way. They concur in the Fermat point. (Proof: exercise, but
recall basic facts about angles in sectors of circles.)

Robin Chapman                           "256 256 256.
Department of Mathematics                O hel, ol rite; 256; whot's
University of Exeter, EX4 4QE, UK        12 tyms 256? Bugird if I no.
rjc@maths.exeter.ac.uk                   2 dificult 2 work out."
http://www.maths.ex.ac.uk/~rjc/rjc.html  Iain M. Banks - Feersum Endjinn

==============================================================================
From: Floor van Lamoen <f.v.lamoen@pi.net>
Newsgroups: sci.math
Subject: Re: Triangle problem
Date: Wed, 22 Oct 1997 16:21:41 -0700

[previous post quoted in entirety -- djr]

Hi,


In addition:
This point is sometimes called Fermat-Torricelli point or even Torricelli 
point. Torricelli showed (1659) that the point P described is the point 
that minimalises AP+BP+CP. This answered a question put forward by 
Fermat.
You could also view 
http://www.evansville.edu/~ck6/tcenters/class/fermat.html on this Fermat 
point.

Regards,
Floor van Lamoen.