I want to find the latitude and longitude of a point, based on knowing three nearby locations, similar to this well written post:

Trilateration using 3 latitude and longitude points, and 3 distances

The quirk in this situation is that I only know relative distances. So for distances to three points (d1,d2,d3) I would only know that:

d1 = 4 * d3 d2 = 2 * d3

Is there a formula that lets me find x1,y1 while minimizing d3? As in, what is the coordinates of the trilateration point where d3 is the smallest value that solves for the equation above?

The additional information that you want the minimum such d3 actually allows us to determine d3 itself if we assume that at least two of these three circles intersect. Consider the circles to have centers (h1,k1), (h2,k2), (h3,k3) and distances from these circles to the desired point to be 4d, 2d and d (d = d3 for brevity). Before we write the equation of circles for solving them, let us check the condition under which those circles intersect each other.

Since, two circles intersect if and only the distance between their centers is at most the sum of their radii, we must have at least one of the following inequalities:

(h1-h2)^2+(k1-k2)^2 <= 4d+2d = 6d
(h2-h3)^2+(k2-k3)^2 <= 2d+d = 3d
(h3-h1)^2+(k3-k1)^2 <= 4d+d = 5d

We can rewrite them as

d >= ((h1-h2)^2+(k1-k2)^2)/6 (say A)
d >= ((h2-h3)^2+(k2-k3)^2)/3 (say B)
d >= ((h3-h1)^2+(k3-k1)^2)/5 (say C)

If all of them hold true, then clearly, d = max(A, B, C). Now we know d and have three equation of circles. This is classic Trilateration. If only the first and second hold true, then d = max(A, B) (other cases are same with cyclic changes).

However, if none of them intersect, then this question is kind of open to what approach we want to take. For example, you could take the centroid (maybe weighted?) of the triangle formed by those three centers.

I hope I understood the question clearly and did not make any typo. P.s. i did not find any option for latex. If there is such an option, feel free to tell me how I can write the equations in latex.

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