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On 22 Jan 2017, at 6:38 AM, Bill Lionheart <NoReply_Lionheart@fer3.com> wrote:

I think this might be a fact that is "well known to those who know".

Suppose you have three non intersection position lines from CN, You
draw the triangle on your chart (so I mean the triangle is small
enough that straight lines on a chart are a good approximation).

Suppose that the errors in the measurements were normally distributed
with the same standard deviation. For example we took enough sextant
readings and averaged and evoked the central limit theorem to say they
should be normal.

So which point do we choose in the triangle as our position? The
"middle"? Well there are 4 well-known centres of triangles (centroid,
orthocentre incentre, circumcentre), and actually there are thousands
of interesting centres of triangle with different properties tabulated
by Kimberling http://mathworld.wolfram.com/KimberlingCenter.html

The Maximum Likelihood estimate (ie in a sense the most likely) is the
one that minimizes the sum of squares of the distances from the
position lines.

This point (number 6 in Kimberling's list!) is called the Symmedian
Point, or Lemoine point or the Grebe point

Diagram here
http://mathworld.wolfram.com/SymmedianPoint.html

Here is "The List" (scroll down to "X(6)")
http://faculty.evansville.edu/ck6/encyclopedia/ETC.html

So that is where you are most likely to be!

Interestingly it can be constructed fairly easily with ruler and
compasses. (Just for fun of course, you would be better off taking the
time to get some more sights if you want better accuracy)

Bill Lionheart