What is the most elegant proof of the Pythagorean theorem?

Question

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield).

What's the most elegant proof?

My favorite is this graphical one:

According to cut-the-knot:

Loomis (pp. 49-50) mentions that the proof "was devised by Maurice Laisnez, a high school boy, in the Junior-Senior High School of South Bend, Ind., and sent to me, May 16, 1939, by his class teacher, Wilson Thornton."

The proof has been published by Rufus Isaac in Mathematics Magazine, Vol. 48 (1975), p. 198.

Answer 1

The proof that uses the fact that shearing a parallelogram parallel to one of its sides preserves area is my favorite. Here's an animation from this site:

Answer 2

This is not really a proof, in the sense that it depends on certain physical assumptions, but it forces you to think very hard about what those physical assumptions are. I learned it from Mark Levi's The Mathematical Mechanic, although unfortunately I don't have a diagram.

Consider a fish tank in the shape of a triangular prism, where the triangle on top has vertices $A, B, C$ ($C$ is the right angle) and the triangle on the bottom has opposite vertices $A', B', C'$. Let $a, b, c$ denote the lengths of the opposite sides. Run a rod through the edge $A A'$ on which the fish tank is allowed to pivot, and fill it with water. By conservation of energy, this system is in mechanical equilibrium.

On the other hand, I claim (and it is not hard to see) that the total torque about the rod is proportional to $c^2 - b^2 - a^2$, each contribution coming from the pressure of the water in the tank against the corresponding side.

Answer 3

More than dissection proofs, I find the proof using similarity most enlightening and intuitive. See post by Terence Tao: drop a perpendicular from C to the hypotenuse AB.

In the figure, the triangles whose are areas are marked x and y are similar to the original triangle (which has area x+y). So accepting that areas of similar right-angled triangles are proportional to the squares of the hypotenuse, x:y:x+y are in ratio a²:b²:c², which is Pythagoras's theorem.

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See also the linked discussion by Alexander Giventhal where he remarks that this proof is more general than tiling or dissection proofs, and is even proved by Euclid. If you think of a²+b²=c² as the geometrical result that the sum of areas of squares constructed with sides a and b is the area of a square placed on c, then the Pythagorean theorem is true not just for constructing squares on the sides, but any similar figures. For instance, as Euclid himself proves, something like the following is true (though it's still just saying a²+b²=c²):

Answer 4

Not a proof in itself, but the book The Pythagorean Proposition by Loomis has probably the most comprehensive collection of proofs of the Pythagorean theorem.

Answer 5

My favorite proof of the Pythagorean Theorem is a special case of this picture-proof of the Law of Cosines:

Drop three perpendiculars and let the definition of cosine give the lengths of the sub-divided segments. Then, observe that like-colored rectangles have the same area (computed in slightly different ways) and the result follows immediately.

When C is a right angle, the blue rectangles vanish and we have the Pythagorean Theorem via what amounts to Proof #5 on Cut-the-Knot's Pythagorean Theorem page. (Note that, as mentioned on CtK, the use of cosine here doesn't amount to an invalid "trigonometric proof".)

The picture works for obtuse C as well. (Proof left as an exercise for the reader.)

A final note ... Because the same-colored rectangles have the same area, they're "equidecomposable" (aka "scissors congruent"): it's possible to cut one into a finite number of polygonal pieces that reassemble to make the other. While there's at least one standard procedure for determining how to make the cuts, the resulting pieces aren't necessarily pretty. Some popular dissection proofs of the Pythagorean Theorem --such as Proof #36 on Cut-the-Knot-- demonstrate a specific, clear pattern for cutting up the figure's three squares, a pattern that applies to all right triangles. I have yet to find a similarly straightforward cutting pattern that would apply to all triangles and show that my same-colored rectangles "obviously" have the same area.

(Another exercise for the reader, perhaps? :)

Answer 6

I really like this one (image taken from cut-the-knot #4)

along with

$$(a+b)^2 = 4\cdot\frac{1}{2}a b + c^2$$ $$\Leftrightarrow c^2 = a^2 + b^2$$

It's so clear and easy ...

Note: Just to make it clear, the side of the square is a+b

Answer 7

http://www.math.ntnu.no/~hanche/pythagoras/

Answer 8

In my opinion Euclid's proof in the Elements, I. 47, is the most brilliant and elegant proof of the pythagorean theorem. One just has to look at the figure to see the WHY of the theorem...the rest is filling in the details. It is just as beautiful and as fresh, and as compelling today, as it was in 300 B.C. (more or less) when Euclid created it.

All mathematicians should see and study the proof at least once in their lives...it is a priceless jewel of mathematical beauty.

Answer 9

Richard Schwartz has a nice Java applet demonstrating a proof by rearrangement of the Pythagorean theorem which is a variation on the one that you like. He constructed a wood model demonstrating this proof, with triangles that swivel.

Answer 10

I think this proof is pretty water tight!

Answer 11

Three similar houses. The red and the blue gables together make up the black one; therefore the same is true for the square side walls.