What are the theorems in mathematics which can be proved using completely different ideas?

One example of a theorem with multiple proofs is the Fundamental Theorem of Algebra (All polynomials in $\mathbb{C}[x]$ have the "right number" of roots). One way to prove this is build up enough complex analysis to prove that every bounded entire function is constant. Another way is to build up algebraic topology and use facts about maps from balls and circles into the punctured plane. Both of these techniques are used specifically to show one such root exists (and once this is proved the rest of the proof is easy).

I think there are other possible proofs of the theorem but these are two I have seen.

The Brouwer fixed-point theorem. The Wikipedia lists the following methods:

• Homology
• Stokes's theorem
• Combinatorial (Sperner's lemma)
• Reducing to the smooth case and using Sard's theorem
• Reducing to the smooth case and using the COV theorem
• Lefschetz fixed-point theorem
• Using Hex