Distance between two vertices is length of shortest path that connects them.
Consider a vertex in a tree, call it V. Pick the (or one of the) vertex in the tree that is farthest from V in terms of distance, call it X. Now pick the vertex (or one of the) vertex in the tree that is farthest from X in terms of distance, call it Y. Show that the shortest path from X to Y is the longest, or one of the longest, shortest path in the tree.
I have a few approaches but they all call for many cases. You can try contradiction after you build your tree, or try rooting your tree at V, but neither of these is that pretty... any innovative approaches?