What Is The Most Elegant Proof Of Pythagoras?

[ToraToraTora]

I love the Euclidian one of course but I'd love to hear about others, possibly Algebraic.....?

[ToraToraTora]

did I put this is the wrong category?

[douglas9401]

As good as any I suppose.

 

[ToraToraTora]

would anyone like to have a go at the OP?

[Buenchico]

I've always been a big fan of proof by contradiction for any mathematical theorem:
https://www.cut-the-knot.org/pythagoras/proof122.shtml

[ToraToraTora]

yep chico, I also love PBCs, such a great way to look it things. 

[Buenchico]

Elliemay1:
That joke has been doing the rounds since it was first published in the Sheffield University Rag Mag in the early 1970s. 

I should know because I was the one who contributed it!

[elliemay1]

Yes, Chris, I remember it from my schooldays!

[tomus42]

?si=YMmj1raWIuwtPAFH

I like this one. Hope the link works.

[ToraToraTora]

Tomus I love that, thanks.

[Gizmonster]

I remember this from school (a LONG time ago), quite a simple proof  :)

Here's a link to the diagram I'll be referring to:

https://i.ibb.co/rtd4Hkp/Pythagorus.png

Consider a square ABCD with a smaller square PQRS within it.

The area of the square ABCD is given by multiplying the lengths of the sides of the square together:

= AB x BC

= (a + b) x (a + b)

= a² + b² + 2ab

This area is also equal to the sum of the areas of the smaller square PQRS and the 4 triangles

= c² + 4 (1/2 x (a x b))

= c² + 2ab

These 2 areas are equal, i.e.

a² + b² + 2ab = c² + 2ab

a² + b² = c²

Q.E.D. 

[ToraToraTora]

cheers giz that's the well known one I was hoping for something different.

[Hymie]

Apparently there are over 350 proofs, but many are empirical (like tomus42’s video of water filled squares).

[ToraToraTora]

Indeed hymie, that's why I am looking for an elegant/amusing one. Thanks for taking time off to contribute.