The number of planar central configurations for the $4$-body problem is finite when $3$ mass positions are fixed

In the $n$-body problem a central configuration is formed when the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. Lindstrom showed for $n=3$ and for $n>4$ that if $n-1$masses are located at fixed points in the plane, then there are only a finite number of ways to position the remaining $n$th mass in such a way that they define a central configuration. Lindstrom leaves open the case $n=4$. In this paper we prove the case $n=4$using as variables the mutual distances between the particles.