Configurations in the Euclidean plane associated to a system of equations

Abstract

In the Euclidean plane ${\bf{E}}^2$, fix four pairwise distinct points \begin{equation*} \label{eqA} \begin{array}{ccc} A=(a_1,a_2),\ B=(b_1,b_2),\ C=(c_1,c_2),\ D=(d_1,d_2), \end{array} \end{equation*} together with four non-zero real numbers $k_A,k_B,k_C,k_D$. We show that System (*) consisting of the following four equations in the unknowns $X=(x_1,x_2)$ and $Y=(y_1,y_2)$ \begin{equation*} \label{egy} \frac{1}{\|X-T\|^2} +\frac{1}{\|Y-T\|^2}=k_T, \quad T\in\{A,B,C,D\} \end{equation*} has finitely many solutions $(X,Y)$ (counting also those with complex coordinates) provided that both of the following two conditions are satisfied: ($i$) no three of the fixed points $A,B,C,D$ are coplanar; ($ii$) no three of the four circles of center $T$ and radius $1/\sqrt{k_T}$ with share a common point in ${\bf{E}}^2$. Furthermore, we exhibit a configuration $ABCD$ showing that System (*) satisfying $(i)$ and $(ii)$ may have many real solutions $(X,Y)$. This result is the planar version of an analog problem in the Euclidean space arising from applications to genetics, investigated in the recent papers \cite{cif} and \cite{ak2024}.