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Please consider the case where I have three points on a 2D plane - $(p_1, p_2, p_3)$ - and a third point, $p_0$, at some unknown position $(x_0,y_0)$. However, I know that $p_0$ is a distance $d_1$ from $p_1$, a distance $d_2$ from $p_2$, and a distance $d_3$ from $p_3$ (all distances are positive real numbers).

What is a closed-form solution for $(x_0,y_0)$ as a function of the distances, $(d_1,d_2,d_3)$ ?

Asking Mathematica to Reduce the expression simply seems to fail.

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If you have two points $p_1$ and $p_2$ with known distances $d_1$ and $d_2$, then the solution is the intersection point of the two circles centered at $p_1$ and $p_2$ with radii $d_1$ and $d_2$ respectively. There are two solutions, in general. With three points, there is generally no solution -- that's why Mathematica is having trouble.

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