Connecting “nearest neighbors.” (A) A set of data points in space. (B) An edge can be formed between $xi$ and $xj$ because there is no other point in the interior of the ball $Bij$ centered halfway between $xi$ and $xj$⁠. (C) Here, because of the presence of a third point $xj$ inside $Bij$⁠, $xi$ and $xj$ cannot be neighbors. (D) Even in the absence of the original data point coordinates (i.e., given only the distances between all pairs of points), Apollonius's formula can be used to determine the length of the segment $p$–$xk$⁠, where $p$ is the center of $Bij$⁠. Namely, $p$–$xk$ is a median of the depicted triangle. Here, because the length of the median is less than the radius of $Bij$⁠, $xi$ and $xj$ cannot be neighbors. (E) Edges are drawn connecting points $xi$ to $xk$ and $xk$ to $xj$ because both $Bik$ and $Bjk$ are empty except for those pairs of points, respectively.