Sperner's lemma (in dimension 2 to keep it visual). The proof in Francis Su's [Monthly paper][1], *Rental harmony: Sperner's lemma in fair division* is especially easy to visualize. Theris a non-empty content, you can have students ponder the role of the hypotheses. And fair division applications allow to motivate it via concrete applications.


  [1]: http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2706