AM-GM Inequality via negative space

This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using algebraic areas and the Side-Angle-Side formula for a parallelogram.

For other visual proofs of this fact see:

For a nearly identical visual proof of another inequality (Cauchy-Schwarz) see

This animation is based on a visual proof that appears in Icons of Mathematics by Claudi Alsina and Roger B. Nelsen (MAA, 2011):

#mathshorts​ #mathvideo​ #math​ #amgminequality #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #calculus #inequality

To learn more about animating with manim, check out:

[MathsScienceandHinduism]

As a math content creator one of my fav visual math channels is yours

[juzbecoz]

This channel is amazing
Now We can IMAGINE MATHS

[error_o]

I can’t help but think of _that one proof_ of the Pythagorean theorem while looking at this

[sanjanapanchal]

Wow that's an interesting way to look at it ❤I am just learning this in my studies and this would be fun to show the teacher for better understanding love ur videos and proofs

[maxwellmogadam]

I would say with the advent of computer visualization for math, it basically changed how I think, I always find myself recreating geometric constructions before writing out the algebra now, which for me is awesome because I also have a love for physics which also benefits from diagrams

[frostyboyee]

My math prof taught us this last week while doing progressions :D nice vid

[tungyeeso]

If two numbers form a fixed sum, their product derived from multiplication is the greatest when the difference between them is zero. The greater the difference between them, the smaller the product you can get. 😊

[FreeGroup]

btw you can show this inequality in 2 lines starting from (sqrt(a)-sqrt(b))²≥0

[Dhanika]

I really need it, the am, gm inequality proof in class 11 is given by an identity (√a+√b) kind of something like this . That I really do not understand
But now it's clear by this visual proof
Thank you @mathvisualproofs

[carl]

sin(x)<=1 => r=tsin(x)<=t with r, t>0. (Had to write it down.)

[xiang-yue-fung]

I still like the version that with a+b diameter circle

[SuperUltraKai]

my brain is gonna already will stop working

[paulh]

(sqrt(a)-sqrt(b))² ≥ 0 therefore sqrt(ab) ≤ (a+b)/2

That's so much shorter, isn't it?

[brd-ch]

Actually i think the proof with a semicircle is much easier and visually better for this

[葉月-dr]

I have a question. Why, after knowing the length of one side of a parallelogram with four identical sides, does its area need to be multiplied by sine to become the area of ​​the circumscribed rectangle?

[blkzz]

Can you please give me a detailed explanation on why you used sin

[Npvsp]

What exactly does it mean “via negative space”? And how is it related to the proof?

[rajkotalovic]

Jos samo da razumem to sto prikazujete

[tunistick]

ooor:
(√a-√b)²≥0
a+b-2√ab≥0
(a+b)/2≥√ab