Factor a Sum of two Squares (visual proof)

This is a short, animated visual proof of showing how to factor a sum of two squares. #manim #math #mathvideo #arithmeticprogression #squares #mathshorts #geometry #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #algebra #pww​​ ​ #proof​ #areas #mathematics​​ #mtbos

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This animation is based on a visual proof from Roger B. Nelsen's second proofs without words compendium.

To learn more about animating with manim, check out:
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Music in this video:
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[mathflipped]

This is a cool formula and a great visualization! I've never seen this formula before. Great job!

[narfharder]

I've never seen this before, it's easy enough to derive if you know what you're after.
x² + y² = (x + y + t)(x + y - t) then solve for t and back-substitute.

I was expecting (x+iy)(x-iy), if you know a way to do a video of that one I'd like to see it.

[jakobthomsen]

I didn't know this one - thanks!
So the sum of two integer squares can be factored into integer factors if both x and y together contain an odd exponent of factor 2 and an even exponent of every other prime factor (so the sqrt(2xy) is an integer). Nice! :D

[columbusmyhw]

Ooh, nice! I've seen this before, but in a slightly different form:
x^4+4y^4 = (x^2+2xy+2y^2)(x^2−2xy+2y^2)
This is Germain's identity, I believe.

[alexanderlin]

Please, also make a video about (a³+b³)=(a+b)×(a²-ab+b²)
and
(a³-b³)=(a-b)×(a²+ab+b²)
Please 🙏
Since some terms are cubed ³, I think some blocks which have 3 dimensions, will need to be animated.
By the way, thank you for this video.

[moamel]

I have think about this formual
a²+b²=(a×b)(2)+(b-a)²

[richardrichardson]

I'm looking for a formula with the value of 2 sqs are known but I want break them out. Say 25 + 49 = 74. So if I start with 74 what is the formula to find both squares please?

[leancaga-anan]

I'm like what the fuck just happened 🤣🤣🤣

[benedettopagano]

How did no one ever teach me this before?

[harshitverma]

It's nothing too unique. Visualization is good though.


x² + y² = (x+y)²-2xy
= (x+y)²- (√2xy)²
Use a²-b² = (a+b)(a-b)

Thus x²+y² = (x+y)²-(√2xy)²
= (x+y+√2xy)(x+y-√2xy)