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Suppose I have an number of distances from an unknown location to a known location. I can use these distances and the known locations to draw a number of circles. The point where all the circles intersect is my unknown location. This is easy to solve. However, if my distance measurements have some error, I will get a cluster of points around my unknown location. The question is, for each pair of intersection points, how can I tell which solution is closer to my unknown point?

Every solution I have come up with involves many special cases, such as when all the known locations are inside one circle, or all the known locations are co-linear, and so on, buy I am hoping to come up with a more elegant solution.

EDIT

Lets make this more simple. I have 2 circles that have at most 2 intersection points. If they don't intersect, I can easily choose the point between them. If they do intersect at 2 points, how can I use a third circle to choose which one of these 2 intersections is the correct one?

My current plan is to do a comparison between the distance between the radius of the third circle and the intersection points and choose the smaller one. The problem with this is that I don't know how much error this approach can tolerate before you choose the wrong side. If I knew that, I could put a goal on my measurement method to try to reduce the error by X%

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    The Wikipedia page for GPS suggests that such a system can be solved using [a generalization of Newton's method](http://en.wikipedia.org/wiki/Newton-Raphson#Nonlinear_systems_of_equations).2010-09-16

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