I like to ask for beautiful mathematical theorems with short proof. A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will know and understand. I do not like to discuss what beauty means...I have now written down a script with 107 such theorems (in my sense) and their proof and I have ideas for another 20. One the one hand I would be very thankful for theorems I have yet not considered. On the other hand I like to see if there is some consensus which theorems with a short proof are beautiful.
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The proof(via the pigeonhole principlecontinued fractions would need too much preparation) that when D>0 is not a square then the "Pellian equation" xxDyy=1 has a nontrivial solution. 


Cantor's diagonal argument to prove that $\mathbb{R}$ is uncountable. 


Euler's formula $$\mathrm{e}^{i \theta} = \cos \left( \theta \right) + i \sin \left( \theta \right)$$ when considered as a theorem. From whatever angle you look at it, almost all the proofs are short and extremely beautiful. 


L.M. Kelly's proof of the SylvesterGallai theorem: in any configuration of $n$ points in the plane, not all on a line, there is a line containing exactly two of the points. See Aigner & Ziegler, "Proofs from the Book", chapter 8. 


@Paul Monsky's proof of Monsky's theorem: a complete proof starting from nothing takes two pages. (doesn't quite meet the criteria, but what the heck). 


The standard evaluation of $\int_{\infty}^{\infty} \exp(x^2) dx.$ 


Fermat's proof, by infinite descent, that there is no Pythagorean right triangle whose area is a square might qualify. 


Picard's little theorem is remarkable, and its oneline proof led Littlewood to remark that it would be the world's shortest Ph.D. thesis. 

