Re: 3-Points of a Triangle by Walter Whiteley

Subject:      Re: 3-Points of a Triangle
Author:       Walter Whiteley <whiteley@mathstat.yorku.ca>
Date:         Sat, 27 Oct 2001 08:05:07 -0400 (EDT)

The issue of 3-points belongs to AFFINE geometry.
Any affine transformation preserves ratios of areas,
to the affine image of a triangle and a 3-point will
be a new triangle with a 3-point of this new triangle.

With that in mind, the only examples you need to study
are 3-points of an equilateral triangle (since this is 
affinely equivalent to any non-collinear triangle).

Some quick investigations suggest that there are other 3-points
for an equilateral triangle.  For example, move one edge in,
parallel, til the line bisects the area.  Do this from a second side.
(Neither of those lines goes through the centroid - it is
not a dilation of root(2) as required for area bisecting,
but is a dilation of 2/3.)

Look at the point of intersection of these two lines.
By symmetry, it must lie on the mirror (median for the
equilateral triangle) from the vertex common to the two 
sides you started with.  Thus it is a 3-point.
By symmetry, there will be such a 3-point on each of the medians.

If you take ANY line bisecting the area of the equilateral triangle,
and a median, and reflect the first line in the median, you should
get a 3-point.  

This does NOT claim all 3-points come that way, but suggests a nice
chunk of the medians are all 3-points.

I wonder whether the entire triangle spanned by these three
vertices is formed of 3-points?  (The centroid of the triangle
is the centroid of this 3-point triangle.)

My previous experience with the Triangle Centers is that these
are defined to the point of being a discrete set (often a unique point).
This is clearly not true here.

Walter Whiteley
York University

> 
> You should visit the Encyclopedia of Triangle Centers.
> 
> Currently, 1114 triangle centers are known.
> 
> http://cedar.evansville.edu/~ck6/encyclopedia/
> 
> --Ed Pegg Jr.
> 
> 
> D & A Klinkenberg wrote:
> 
> > Hello gdb and all readers,
> > 
> > My quick answer is 0%.   (I have been thinking of another centroid problem.)
> > I thought that all lines through the centroid, divide a triangle (and other
> > plane figures) into two equal areas.  I also thought that all lines which
> > divide the triangle into two equal areas, passes through the centroid.
> > 
> > However, I thought a triangle had several 3-points besides the centroid.
> > Gerald, do you also have other 3-points in mind?  If so, would you please
> > list the others to consider for your question?
> > 
> > Dan in NY
> > 
> > &&&        "Gerald Brown" <gdb8b@hotmail.com> wrote in message
> > news://4g19j7ign04e@legacy.../
> > 
> >>   Here is a problem that I have been working on for awhile:
> >>
> >>   A 3-point of a triangle is defined as a point where there exists
> >>three distinct lines through the point and each line divides the
> >>triangle into two polygons of equal area. Clearly, the centroid of the
> >>triangle is a 3-point of the triangle. What percent of the area of a
> >>triangle is the area of the set of 3-points of the triangle?
> >>
> > gdb
> > 
>