Followup to Favorite Proof

Results to Favorite Proof:

What's your name/twitter?Name of proofLink to proof somewhere online...Why do you like this proof?Additional testaments of love for this proofAdditional testaments of love for this proof...          
@dandersodEuclid's Proof of Infinte PrimeslinkLove proof by contradiction. Simple, powerful, easy to understand.From the book
@sophgermainWhy square root of 2 is irrationallinkIt's the first proof i show students and it's simple.
@nik_d_mathsInduction for sum of squareslinkProof by induction is fun, and this involves enough algebra for students to feel they have done something, while being short enough to use early onI love induction?
@stwwilkinsonUncountability of the real numbers (Cantor)linkvery simple to explain, yet challenges a lot of intuitive notions about infinityThe set up for this proof -- the countability of the rational numbers -- is also mind blowing.
@MrHonnerMidline of a TrianglelinkElementary, but powerful idea in geometry; several different ways to think about why it's trueGreat way to demonstrate the power of coordinate geometry for proofLeads to beautiful results like Varignon's Theorem
@Moko58Product of 2 odd number is an odd numberlinkEven beginners can attempt this proof.this is a great proof.
@mpershanIncompleteness of axioms of arithmeticlinkIt's problably the hardest mathematical idea that I even sort of understand, which is definitely part of why I like it, but the self-referential sentence is brilliant and all sorts of cool things follo wform it.
@mathymcmathersoGeometric Proof of the Pythagorean TheoremlinkHas a visual and algebraic component that go hand-in-hand. Personally: most convincing and direct proof of pythagorean theorem I've ever seen
@j_lanierThere exists an irrational number that when raised to an irrational power results in a rational number.linkI like this proof because it is simple; it yields a positive result, rather than a "there exists no..."; and it's concrete but so delightfully non-constructive--we find out that there exists such a thing without having a single example of it! So cheeky.I can't promise that it's my favorite ever, but it's a dang neat proof.
@DanielPearcyCan't pick between irrationality of root(2), infinitely many primes, uncountability of realsLooks like I'm a sucker for proof by contradiction.Funnily enough I don't know many recent (last 100 years) proofs

A pretty common theme: Simplicity in presentation or Visual Component.

<snark>Note: No 2 column geometry proofs are found in this list.</snark>

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