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Re: 3-Points of a Triangle
Posted:
Dec 9, 2001 9:46 AM
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My result is that the relative area covered by 3-points
in an arbitrary triangle is:
3/4 Ln 2 - 1/2 = 0.0198604.
Can anybody check it?!
ZFS
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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
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