Rewriting this equation without the square roots? Given the following system of equations:
$\sqrt{(x-x_1)^2+(y-y_1)^2}+s(t_2-t_1) = \sqrt{(x-x_2)^2 + (y-y_2)^2}$
$\sqrt{(x-x_2)^2+(y-y_2)^2}+s(t_3-t_2) = \sqrt{(x-x_3)^2 + (y-y_3)^2}$
$\sqrt{(x-x_3)^2+(y-y_3)^2}+s(t_3-t_1) = \sqrt{(x-x_1)^2 + (y-y_1)^2}$
How could I write this without square roots? Here, the goal is to solve the system to determine the unknowns $x$ and $y$. How do I solve this (the goal is to do so algorithmically)?

$$
\sqrt{a^2}+b = \sqrt{c^2} \\
\implies a^2 + b^2 + 2b\sqrt{a^2} = c^2 \\
\implies a^2 + b^2 + 2b(\sqrt{c^2}-b) = c^2 \\
\implies \sqrt{c^2} = \frac{c^2 -a^2 - b^2}{2b} + b \\
\implies c^2 = \left(\frac{c^2 -a^2 - b^2}{2b} + b \right)^2
$$
$$
\iff c = \pm \left(\frac{c^2 -a^2 - b^2}{2b} + b \right)
$$
