Finding location of a point on 2D plane, given the distances to three other know points I need to find location of the point $s_0$; the locations of other three points ($s_1$, $s_2$, $s_3$) are known. $d_i$ are known distances.
Given: $x_1$, $x_2$, $x_3$, $y_1$, $y_2$, $y_3$, $d_1$, $d_2$, $d_3$
To be found: $x_0$, $y_0$
This will be a physical system. $s_i$ will be fixed antennas and the distances $d_i$ will be read via proximity sensors. The aim will be to make $s_0$ find its own location. So, $s_0$ will always exist; there will always be a unique solution point $s_0:(x_0, y_0)$.
How do I solve this problem?


essentially you are solving the following equation in $x_0$ and $y_0$:
$$\begin{cases}(x_0-x_1)^2+(y_0-y_1)^2=d_1^2 \\
(x_0-x_2)^2+(y_0-y_2)^2=d_2^2 \\
(x_0-x_3)^2+(y_0-y_3)^2=d_3^2\end{cases}$$
solving only the first 2 will give you 0, 1 or 2 solutions for which you can select the correct one from the third equation
with the formulas taken from http://paulbourke.net/geometry/2circle/
if $|S_1S_2|> d_1+d_2$ or $|S_1S_2|< |d_1-d_2|$ then there are no solutions and if $S_1=S_2$ and $d_1+d_2$ then all the points on the entire circle are solutions
and with $a=\frac{d_1^2 - d_2^2 + |S_1S_2|^2 }{ 2 *|S_1S_2| }$ and $h^2 = d_1^2-a^2$we have
$$\begin{cases}x_0 = x_1 \pm h*\frac{ y_2 - y_1 }{|S_1S_2|}\\
y_0 = y_1 \mp h*\frac{ x_2 - x_1 }{|S_1S_2|}\end{cases}$$
as possible solutions insert these into the third equation and eliminate the one that doesn't fit
