Fréchet barycenter on circle

Question

Let $(x_i)$ be $n$ points a the circle. Let $F(x)=\sum_i d(x,x_i)^2$ be a function on the circle. When it exists, let $B(x_1,\ldots,x_n)=\operatorname{argmin}(F)$. $B$ is called the Fréchet barycenter. When things are well defined, is the Fréchet barycenter associative? I can't find a counter example.