Date: Tue, 21 Oct 1997 07:51:37 -0600 From: Robin Chapman Subject: Re: Triangle problem Newsgroups: sci.math In article <62grbc\$rod@eng-ser1.erg.cuhk.edu.hk>, Peter Tam 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 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.