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Slightly as an aside here are a couple of Geogebra animations people
may like to play with

In both the triangle is A,B,C and you can drag the vertices of the
triangle around to see what happens

This one plots an example of a least squares ellipse and the axes. The
points labelled X1344 an X1345 are Kimberling triangle centres that
lie on the major and minor axes of the ellipse. F1 and F2 are the foci
of the ellipse
https://www.geogebra.org/classic/jtc8czqb

In this one the Orthocentre H and the Circumcentre O of the triangle
are plotted. But if you look on the left you will see three
interesting parameters J=|OH|/R  where R is the circumradius. Q the
sum of squares of sines of interior angles, which was mentioned by
Robin in a post he quoted on this thread, and the condition number of
the system of equations for the fix which is 1 for equilateral (best
conditioned system) , Between 1 and two 2 for acute and more than 2
for  obtuse.  Not J goes from zero to 3.  Drag around one vertex and
go from equilateral through right angle to acute and see how they
change.

https://www.geogebra.org/classic/ytk42cnd


Bill

On Sat, 23 Mar 2019 at 16:33, Robin Stuart  wrote:
>
> Bill,
>
>      My point (which is somewhat buried in this post) is that from a purely 
statistical point of view the obtuse 0°, 60°, 120° triangle is equally as 
good as the equilateral 0°, 120°, 240° one since they both constrain the 
observer's position equally well. The equilateral triangle wins in the 
presence of a systematic error like I.E. and any empirical bias toward 
against obtuse triangles has that as a justification,
>
> Regards,
>
> Robin
>
> 

   

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