Is it any clearer to you now?

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Is it any clearer to you now?

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Draw a square in a piece of paper. Bend that piece of paper. OMFG a 2D figure projected in 3D curved space! Holy sh*t that must be the work of the devil for you.

Your picture only demonstrates that it is true in 2D Euclidean space. Good luck demonstrating that the areas remain the same when the square lives in curved space, here`s a hint: they don`t. Ergo, your drawing doesn`t say WHY, by any chance, the sum of the areas MUST be unequivocally the same. Believe it or not, questions like that keep us mathematicians up at night. No meds necessary though.

Stay off those drugs, dude. Hint: a *square* (which Pythagoras` Theorem makes heavy use of) is a two-dimensional shape. It only exists in Euclidean space. Otherwise it`d be a cube. Or a tesseract. Or an n-dimensional hypercube. The diagram below demonstrates exactly why Pythagoras` Theorem works - namely that exactly the same amount of space is left over when you surround a square of side length equal to the hypotenuse or two squares with side lengths equal to the lengths of the opposite and ajacent sides laid corner to corner with a larger square of the same size in each case. It is not a "matter of research", it`s a mathematical proof that`s been fully understood for thousands of years.

Why it`s true is still a subject of research, as is my understanding, since it doesn`t have to be true (and in fact it isn`t in all kinds of spaces, it is only true in flat/euclidean space)

@Big61AL: Sure, what you have to measure is the area though, so you have to cut it up in squares and fit them all in the areas of the other two squares. Or you could just flow the area from one to the other two and let fluid mechanics do the arranging.

@carmium: it`s actually geometry, but it`s slightly more complicated than just squaring the sides. It`s not immediately obvious that the area of the bigger square is exactly the sum of the area of the other two squares. Just like it`s not obvious that the internal angles all add up to 180 degrees (in Euclidean spaces)

(A lot of work just for people who don`t trust arithmetic.)

- Is it any clearer to you now?