Is the restriction on this system necessary? I'm trying to solve the following non-linear system:
$\sqrt{(x-x_1)^2+(y-y_1)^2}+s(t_2-t_1) = \sqrt{(x-x_2)^2 + (y-y_2)^2}$
$\sqrt{(x-x_2)^2+(y-y_2)^2}+s(t_3-t_2) = \sqrt{(x-x_3)^2 + (y-y_3)^2}$
$\sqrt{(x-x_3)^2+(y-y_3)^2}+s(t_1-t_3) = \sqrt{(x-x_1)^2 + (y-y_1)^2}$
For the unknowns, $x$ and $y$. The system is the solution to a localization problem, where, given the coordinates $[x_i,y_i]$ of three parties, the time at which each party "saw" some signal ($t_i$), and the speed of that signal, $s$, the coordinates of the source are given by $[x,y]$. Here, we assume that the coordinates and the source are coplanar.
I read somewhere that, to write the system above, one must have that $t_1 < t_2 < t_3$. Is this the case? And if so, why?
My goal right now is to solve for $[x,y]$ algorithmically, by setting the equations equal to 0 and plugging them into a root-finder. I want to do this hundreds of times, so as to localize hundreds of events, and in many cases this constraint is not met. Can I still simply plug the values in?

The restriction that $t_1 < t_2 < t_3$ need not be met to solve these equations and get a meaningful answer. If the restriction were necessary, this still wouldn't be a problem from an "algorithmic" point of view. We could always just rename the observers in an order aligned with $t_1 < t_2 < t_3 \; .$ In other words we would arbitrarily state that the observer that saw the signal first has coordinates $(x_1, y_1) \;$ and continue on from there.
Once again though, the equations can be solved no matter which numbered observer sees the signal first. This is true in 3d as well.
