Centroid sum-of-squares identity (${\sum }_i\Vert x_i-z{\Vert }^2={\sum }_i\Vert x_i-\bar{x}{\Vert }^2+n\Vert z-\bar{x}{\Vert }^2$)

For a nonempty finite collection of Euclidean vectors, its sample mean $\bar{x}=n^{-1}{\sum }_i{x}_i$ uniquely minimizes the sum of squared Euclidean distances to a proposed centroid $z$. Expand each squared Euclidean norm around $\bar{x}$ and use ${\sum }_i(x_i-\bar{x})=0$ to cancel the cross term. The displayed identity leaves a nonnegative term $n\Vert z-\bar{x}{\Vert }^2$, equal to zero exactly at $z=\bar{x}$.