Prove the following claim:

In an undirected and unweighted graph with k vertices, If the length of the shortest path between vertices u and v is greater than k/2 then there must be a vertex w not equal to u or v whose removal disconnects u and v.

I want to do some kind of proof involving the Pigeonhole Principle to show that multiple paths between u and v must share at least one vertex (whose removal would disconnect u and v), but I can't seem to make it work. Any ideas?