= Solution
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 $BC$:
$$
d(Q,P)\le\max\{d(Q,A),d(Q,w)\}
\le\max\{d(Q,A),d(Q,B),d(Q,C)\}.
$$
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 \b[the <diameter> is exactly the longest side length]:
$$
\boxed{\operatorname{diam}(ABC)=\max\{d(A,B),d(B,C),d(C,A)\}.}
$$
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