Visual Proof of AM-GM Inequality I

This is a short, animated visual proof of the arithmetic mean-geometric mean inequality using areas. This theorem states that the average of two positive numbers is greater than or equal to the square root of the product of the same two numbers (with equality only when the two numbers are the same). #mathshorts #mathvideo #math #amgminequality #area #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof

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Here are some other visual proof of the AMGM inequality (and others):

To learn more about animating with manim, check out:

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Background music, used according to CC BY-SA 3.0, is "The Road" from Esteban Orlando:

Esteban Orlando:

[VENKATAMITHWOONNABCE]

Nice way to prove the AM-GM Inequality using visual representation and the magic of manim made it much more beautiful sir

[QingxiangJia]

Intuitive and elegant. Dr. Edgar is the best Math professor!

[Luizabf]

A prova mais elementar, a melhor de todas

[Grateful.For.Everything]

I’m thinking that white area can be seen as reserved space from which the rest of it all pivots around, so that it has an equalized axis that balances the symmetry out.

[DitDede]

This is basically a visualization of the algebraic proof. The semicircle one is the geometric proof.
Challenge: make a video with a visual proof for the general theorem, when averaging n terms :-)

[terencekuhl]

I don't quite understand how the equality sign in 4xy = (x + y)^2 - (y-x)^2 turns into a less or equal sign.
My assumption is that since the right side is non negative but the left sight can be negative it is therefore smaller but I'm not sure, but could anyone please further explain why it is non negative and if my assumption is right?