While solving a bigger problem, I stumbeled upon a system of parametric equations $$ \left\{ \begin{array}{ll} \dfrac{x-a}{\sqrt{\left(x-a\right)^2+\left(y-b\right)^2}} + \dfrac{x-c}{\sqrt{\left(x-c\right)^2+\left(y-d\right)^2}} = 0\\ \dfrac{y-b}{\sqrt{\left(x-a\right)^2+\left(y-b\right)^2}} + \dfrac{y-d}{\sqrt{\left(x-c\right)^2+\left(y-d\right)^2}} = 0 \end{array} \right. $$ I need to solve it. I don't actually need all of the solutions to this system, one is enough.
Thanks, in advance.
Note that $d_1=\sqrt{\left(x-a\right)^2+\left(y-b\right)^2}$ and $d_2=\sqrt{\left(x-c\right)^2+\left(y-d\right)^2}$ are the distances from the point $(x,y)$ to the points $(a,b)$ and $(c,d)$, respectively. Rewrite the given system as
$$\frac{x-a}{c-x}= \frac{y-b}{d-y}= \frac{d_1}{d_2}$$
So, geometrically, $(x,y)$ represent all the points on the line connecting $(a,b)$ and $(c,d)$.
This seems to be : find the extremum of function $$f(x,y)=\sqrt{(a-x)^2+(b-y)^2}+\sqrt{(c-x)^2+(d-y)^2}$$ If you do not have any constraint, it could be any point along the line explained by @Quanto in his/her answer.
