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On 2010-12-09 20:25, John Karl wrote:
> BTW, attached is a case which clearly shows that neither the Fermat
> point, the center of gravity, the bisecting of the triangles vertices,
> or the Steiner point is located at the maximum probability (marked
> with an "x").

Why, of course the MPP is at the symmedian point! In one form or
another, this has been known in French circles since Villarceau's
seminal publication on Nouvelle navigation in 1877.  When solving the
minimum condition by partial differentiation, Villarceau found the
solution to be at the point for which the distances to the sides of the
triangle are as the sides of the triangle themselves. (In other words
x:y:z = a:b:c) This is a defining property of the symmedian point.
Lemoine showed independently, but around the same time, that the
symmedian point has the property of minimizing the square sum of the
distances.

The geometrical construction of the symmedian point based on the above
proportion is so simple that one must wonder why generations of
navigation textbooks including Dutton, Bowdtich, etc have been
speculating for over a century about various construction methods (like
the ones that Karl mentions) without ever hitting on the only correct one.

Herbert Prinz

   

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