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Let $$d: \mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{R}$$ be defined by $$d(x_i,x_j)=\frac{|x_i-x_j|}{\sqrt{M(i)M(j)}},$$ where $M(i)$ represents the average distance between $x_i$ and the other points, $M(j)$ represents the average distance between $x_j$ and the other points, and $|x_i-x_j|$ is the standard Euclidean distance. I need to prove that $d$ satisfies the triangle inequality. Thanks!

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