### Three position lines don't meet. Where are you?

I think this might be a fact that is "well know to those who know". I often give an example when teaching inverse problems of three equations in two unknowns, and we calculate the least squares solution using the Moore-Penrose generalized inverse.

I also give an example from navigation. Suppose you have three non intersection position lines and you draw them on your chart (map). You got these by measuring the angle between stars and the horizon. They don't exactly meet as you have errors in your measurements.

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"? The Admiralty Manual of navigation suggests choosing the corner of the triangle closest to danger. But if we are way out in a featureless sea we just want to choose the best.

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

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 Kimberlings list! ) is called the Symmedian Point, or Lemoine point or the Grebe point

Diagram on Wolfram

Here is "Kimberling's List" (scroll down to "X(6)")

So that is where you are "most likely" to be!

Interestingly it can be constructed fairly easily with ruler and compasses.

I also give an example from navigation. Suppose you have three non intersection position lines and you draw them on your chart (map). You got these by measuring the angle between stars and the horizon. They don't exactly meet as you have errors in your measurements.

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"? The Admiralty Manual of navigation suggests choosing the corner of the triangle closest to danger. But if we are way out in a featureless sea we just want to choose the best.

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

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 Kimberlings list! ) is called the Symmedian Point, or Lemoine point or the Grebe point

Diagram on Wolfram

Here is "Kimberling's List" (scroll down to "X(6)")

So that is where you are "most likely" to be!

Interestingly it can be constructed fairly easily with ruler and compasses.