There are thousands of different proofs of the Pythagorean theorem, and some of them are really cool. The purely trigonometric proof that was found by some high school students recently is a great one. However, I think the greatest proof of all is this little gem that has been attributed to Einstein [1].

Take any right triangle. You can divide it into two non-overlapping right triangles that are both similar to the original triangle by dropping a perpendicular from the right angle to the hypotenuse. To see that the triangles are similar, you just compare interior angles. (It's better to leave that as an exercise than to describe it in words, but in any case, this is a very commonly known construction.) The areas of the two small triangles add up to the area of the big triangle, but the two small triangles have the two legs of the big triangle as their respective hypotenuses. Because area scales as the square of the similarity ratio (which I think is intuitively obvious), it follows that the squares of the legs' lengths must add up to the square of the hypotenuse's length, QED.

It's really a perfect proof: it's simple, intuitive, as direct as possible, and it's pretty much impossible to forget.

[1] https://paradise.caltech.edu/ist4/lectures/Einstein%E2%80%99...

I must confess I clicked through hoping to see a comic of Garfield the cat using pizza slices to approximate right triangles.

American presidents used to be smart.

I was ready for it to involve lasagna.

Not to bash the former president, but I'm failing to see what's so clever or nice about the proof... could someone please explain if I'm missing something? If you're going solve it with algebra on top of the similar triangles and geometry anyway, why complicate it so much? Why not just drop the height h and be done with it? You have 2 a b = 2 c h, c1/a = h/b, c2/b = h/a, c = c1 + c2, so just solve for h and c1 and c2 and simplify. So why would you go through the trouble of introducing an extra point outside the diagram, drawing an extra triangle, proving that you get a trapezoid, assuming you know the formula for the area of a trapezoid, then solving the resulting equations...? Is there any advantage at all to doing this? It seems to make strictly more assumptions and be strictly more complicated, and it doesn't seem to be any easier to see, or to convey any sort of new intuition... does it?

He was also the 20th president of the USA.

Netflix recently released a mini-series on Garfield's election and presidency: https://www.themoviedb.org/tv/245219-death-by-lightning.

AFAIK: Pythagoras never wrote about this Triangle Theorem. There's no proof that he ever even knew about it. But he had mandated to his Pythagorean school (students) that any discovery or invention they made would be attributed to him instead.

The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch.

Interestingly: the Triangle Theorem was discovered, known and used by the ancient Indians and ancient Babylonians & Egyptians long before the ancient Greeks came to know about it. India's ancient temples are built using this theorem, India's mathematician Boudhyana (c. ~800 BCE) wrote about it in his Baudhayana Shulba (Shulva) Sutras around 800 BCE, the Egyptian pharoahs built the pyramids using this triangle theorem.

Baudhāyana, (fl. c. 800 BCE) was the author of the Baudhayana sūtras, which cover dharma, daily ritual, mathematics, etc. He belongs to the Yajurveda school, and is older than the other sūtra author Āpastambha. He was the author of the earliest Sulba Sūtra—appendices to the Vedas giving rules for the construction of altars—called the Baudhāyana Śulbasûtra. These are notable from the point of view of mathematics, for containing several important mathematical results, including giving a value of pi to some degree of precision, and stating a version of what is now known as the Pythagorean theorem. Source: http://en.wikipedia.org/wiki/Baudhayana

Baudhyana lived and wrote such incredible mathematical insights several centuries before Pythagoras.

Note that Baudhayana Shulba Sutra not only gives a statement of the Triangle Theorem, it also gives proof of it.

There is a difference between discovering Pythagorean triplets (ex 6:8:10) and proving the Pythagorean theorem (a2 + b2 = c2 ). Ancient Babylonians accomplished only the former, whereas ancient Indians accomplished both. Specifically, Baudhayana gives a geometrical proof of the triangle theorem for an isosceles right triangle.

The four major Shulba Sutras, which are mathematically the most significant, are those attributed to Baudhayana, Manava, Apastamba and Katyayana.

Refer to: Boyer, Carl B. (1991). A History of Mathematics (Second ed.), John Wiley & Sons. ISBN 0-471-54397-7. Boyer (1991), p. 207, says: "We find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. However all of these triads are easily derived from the old Babylonian rule; hence, Mesopotamian influence in the Sulvasutras is not unlikely. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides, but this form of the Pythagorean theorem also may have been derived from Mesopotamia. ... So conjectural are the origin and period of the Sulbasutras that we cannot tell whether or not the rules are related to early Egyptian surveying or to the later Greek problem of altar doubling. They are variously dated within an interval of almost a thousand years stretching from the eighth century B.C. to the second century of our era."

That looks like "half" of the proof using a square:

https://www.onlinemathlearning.com/image-files/xpythagorean-...

where you draw three extra triangles, not just one, and they surround a square of c x c. Think about it as making two copies of the trapezoid, one rotated on top of the other.