How I can solve the system of equations below? [closed]

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$$\begin{cases} \sqrt{(x - X_1)^2+(y - Y_1)^2} = C1 \\ \sqrt{(x - X_2)^2+(y - Y_2)^2} = C2 \\ \sqrt{(x - X_3)^2+(y - Y_3)^2} = C3 \\ \end{cases} $$

$C1, C2, C3, X_1, Y_1, X_2, Y_2, X_3, Y_3 \in R$
$C1, C2, C3, X_1, Y_1, X_2, Y_2, X_3, Y_3$ are constant

How I can calculate $(x, y)$?

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2 Answers

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Each one of the three equations describes a circle, and the system requires the point $(x, y)$ to be on all three circles.

So one quick way to solve this system is look at the first two equations, which yield two possible solutions (Two circles coincide at no more than two points). Then check for the two solutions whether they satisfy the third equation in the system, and let $(x, y)$ the one that does. Note that there is a chance that the solution does not exists.

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you should square each equation, then solve it like a linear system, by combination or substitution methods to find $x^2$, $y^2$ and $z^2$, and then square root each of them.

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