Beautiful, simple proofs worthy of writing on this beautiful glass door

Barak beat me to my #1 choice. This would be second:

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$\qquad\qquad\qquad\qquad\quad$ Geometric Explanation of the Binomial Theorem

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$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ Proof that $~\displaystyle\sum_{k=1}^n(2k-1)=n^2$

Cosines and Sines Around the Unit Circle

Trigonometric Angle Sum and Difference