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...

This then becomes a restatement of another classic proof (the simple algebraic proof given near the top of the main Wikipedia page for the theorem). So we can imagine Garfield discovering this approach by cutting that diagram (https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/Fil...) in half and describing a different way to construct it.

He was assassinated early and barely got to serve. The story of his life, the shooting, and the subsequent medical drama (featuring even a cameo by Alexander Graham Bell improvising a diagnostic device) are so epic you have to wonder if time travelers are messing with us.

His legacy was the nonpartisan professional civil service, a key part of his agenda that his successor felt obligated to carry out, an accomplishment that recently came under particularly heavy attack.

Netflix just came out with the miniseries about him, 'Death by Lightning,' based on the book 'Destiny of the Republic.' His earlier life is featured prominently in '1861: The Civil War Awakening' by Adam Goodheart. There are a few great C-SPAN/Book TV videos by some of the authors that tell the story concisely and convey why some of us are so fascinated by that history.

This produces an inner square’s edge with sides length c and four equal right triangles of sides a, b, and c.

Note that the area of the outer square equals the sum of the inner square plus the area of the four triangles. Solve this equality.

> From the figure, one can easily see that the triangles ABC and BDE are congruent.

I must confess I do not easily see this. It's been a long time since I did any geometry, could someone help me out? I'm probably forgetting some trivial fact about triangles.

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."

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.

https://www.youtube.com/watch?v=QZWS2g-fEAU