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. 

*

*$20$ different proofs 0f the Euler Formula
$$\color{Red}{V-E+F=2.}$$

*$10$ different proofs 0f the  Gaussian Integral
$$\color{Green}{\int_{\mathbb{R}}e^{-x^2}\,\mathrm{d}x=\sqrt\pi.}$$
