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A subset of a Riemannian manifold is geodesically convex when any two of its points can be joined by a minimizing geodesic segment lying in the subset. In the hyperbolic plane the joining geodesic segment is unique. Thus in the Beltrami-Klein model, geodesically convex subsets correspond to Euclidean convex sets.

Consider the filled hyperbolic triangle with vertices $A,B,C$. Every point $P$ is on some geodesic segment $Aw$ with $w\in BC$: for instance, in the Beltrami-Klein model these are the ordinary straight segments filling a convex triangle. For any other point $Q$ of the triangle, apply part (b) first on $Aw$ and then on $BC$:Applying the same argument to $Q$, with each vertex held fixed, bounds every term on the right by the maximum of the three side lengths. The endpoints of a longest side attain that bound. Thus the diameter is exactly the longest side length: