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By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I can find the usual proofs on the internet but I was wondering if someone knew a proof that is unexpected in some way. Induction, backward induction, use of Jensen inequality, swapping terms, use of Lagrange multiplier, proof using thermodynamics yeah, I know, it's rather some physical argument that this theorem might be true, not really a proof , convexity, … are some of the proofs I know. I shall provide a simple geometric proof of the inequality in the case of two variables which I have not been able to find anywhere else - a proof involving a triangle in a circle seems to be popular. This proves the two-variable case. Inequality for all

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Inequality for all

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Inequality for all

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Inequality for all

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