The butterfly theorem is notoriously tricky to prove using only "high-school geometry" but it can be proved elegantly once you think in terms of projective geometry, as explained in Ruelle's book The Mathematician's Brain or Shifman's book You Failed Your Math Test, Comrade Einstein.

Are there other good examples of simply stated theorems in Euclidean geometry that have surprising, elegant proofs using more advanced concepts? Such examples are valuable pedagogically since they illustrate the power of the advanced methods.