What are the theorems in mathematics which can be proved using completely different ideas? I would like to know about theorems  which can give different proofs using completely different techniques.  

Motivation:
  When I read from the book Proof from the Book, I saw there were many many proof for the same theorem using completely different fields of mathematics.
  (There were Six proofs of the infinity of primes based on different ideas even using topology.)

I wonder How mathematicians give this kind of proofs.
If you know such theorems and proofs please share them with me. Thank you. 

As a kind of extreme example, I've heard of this book, reputed to have given over 360 proofs of the Pythagorean theorem. I'm not sure how pairwise different they are, but there's bound to be a lot of variety.
Another one that sticks in my mind is the proof of the Cayley-Hamilton theorem 
 for real (or complex) matrices. There is an algebraic proof that works for any field at all, but for the reals and complexes, you can argue topologically that the matrix you are working with is the limit of a sequence of diagonalizable matrices, and since the approximations satisfy their characteristic polynomial, so does the limit.
