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Let $X$ consist of points on the surface of a sphere. Define a distance function $d(P,Q)$ as the length of the arc of the great circle passing through points $P$ and $Q$ on the surface of the sphere. Show that $d$ satisfies the triangle inequality.

I'm not sure how to get started. One way could be to express the arc lengths (say between $P$ and $Q$) in terms of the angle between the lines joining the center of the sphere to $P$ and $Q$. So if $\theta_1$ is the angle between $P$ and $Q$, $\theta_2$ is the angle between $Q$ and $R$, and $\theta_3$ is the angle between $P$ and $R$, then we'd effectively have to prove that $\theta_i + \theta_j \geq \theta_k$.

Beyond this, I'm at a loss on how to proceed, or whether this is even the correct method to go about.

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Perhaps this is not so helpful but your 'distance' is the length of a geodesic curve (with respect to the Riemannian metric induced by the Euclidean one on $\mathbb{R}^n$). For example, you can see this curve as being of the form $$\gamma(t) = \cos (t) P + \sin (t) Q' $$ where $P, Q'$ are on the sphere, are orthogonal, and span the same plane as your initial points $P,Q$.

Since the sphere is compact, the lengths of these geodesic paths (and hence your 'distance') coincide with the Riemannian distance function on the sphere.

A reference is Lee's book on Riemannian manifolds.

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