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    Re: Where are you most likely to be in the triangle?
    From: Stan K
    Date: 2017 Jan 21, 18:17 -0500
    Which one of these, if any, corresponds to the point that Herbert Prinz discussed at a Mystic Seaport Navigation Weekend a bunch of years ago?


    -----Original Message-----
    From: Bill Lionheart <NoReply_Lionheart@fer3.com>
    To: slk1000 <slk1000@aol.com>
    Sent: Sat, Jan 21, 2017 5:39 pm
    Subject: [NavList] Where are you most likely to be in the triangle?

    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
    
    
       
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