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Here is a proof of Pythagoras's theorem that I like:

• You want to prove that the sum of the squares on each of the non-hypotenuse sides equals the square on the hypotenuse.

• You generalize, and instead prove that for any shape, if you scale it by $a$, and then by $b$, the sum of the resulting areas is the area of the shape scaled by $c$. (We began with the case of the unit square.)

• By thinking about how areas scale, it suffices to check for one particular shape.

• We check it by taking the shape to be the original triangle (to be pedantic: scaled so that its hypotenuse has length one). This case is clear: just drop a perpendicular from the vertex opposite the hypotenuse to the hypotenuse, and see note that the triangle with hypotenuse length $c$ is the sum of two similar triangle of hypotenuse lengths $a$ and $b$.

Obviously you are not going to drop this into casual conversation: it requires a focused effort at explanation. But I think it illustrates something true, and fairly general, about how mathematicians argue.

For example, the squares that we are adding get transmuted from areas of specific shapes, to scaling factors for areas of quite general shapes. This lets us go from checking something tricky (or trying to make tricky geometric constructions with the squares on the three sides, to see how they are related) to checking something that is immediately obvious. So we also see two kinds of ingenuity: the ingenious reinterpretation of the meaning of the squares (ingenious on a conceptual level, which is certainly a very common form of mathematical ingenuity), and the ingenuity of drawing a single line on the original triangle to break it into two triangles similar to itself (which is ingenious on a more visceral level --- a single stroke of ingenuity).

One other advantage: the other participants quite likely have heard of Pythagoras's theorem, and may even remember it, but are unlikely to know a proof. (Or do painters have some training in Euclidean geometry?) Knowing the result, they may be better positioned to appreciate the ingenuity of the proof.

Also: this argument appeared somewhere else recently on MO, I think, but I forget where. (Added: here, in a reponse of Dick Palais to Timothy Gowers's question about making something easier by generalization.)

Here is a proof of Pythagoras's theorem that I like:

• You want to prove that the sum of the squares on each of the non-hypotenuse sides equals the square on the hypotenuse.

• You generalize, and instead prove that for any shape, if you scale it by $a$, and then by $b$, the sum of the resulting areas is the area of the shape scaled by $c$. (We began with the case of the unit square.)

• By thinking about how areas scale, it suffices to check for one particular shape.

• We check it by taking the shape to be the original triangle (to be pedantic: scaled so that its hypotenuse has length one). This case is clear: just drop a perpendicular from the vertex opposite the hypotenuse to the hypotenuse, and see note that the triangle with hypotenuse length $c$ is the sum of two similar triangle of hypotenuse lengths $a$ and $b$.

Obviously you are not going to drop this into casual conversation: it requires a focused effort at explanation. But I think it illustrates something true, and fairly general, about how mathematicians argue.

For example, the squares that we are adding get transmuted from areas of specific shapes, to scaling factors for areas of quite general shapes. This lets us go from checking something tricky (or trying to make tricky geometric constructions with the squares on the three sides, to see how they are related) to checking something that is immediately obvious. So we also see two kinds of ingenuity: the ingenious reinterpretation of the meaning of the squares (ingenious on a conceptual level, which is certainly a very common form of mathematical ingenuity), and the ingenuity of drawing a single line on the original triangle to break into two triangles similar to itself (which is ingenious on a more visceral level --- a single stroke of ingenuity).

One other advantage: the other participants quite likely have heard of Pythagoras's theorem, and may even remember it, but are unlikely to know a proof. (Or do painters have some training in Euclidean geometry?) Knowing the result, they may be better positioned to appreciate the ingenuity of the proof.

Also: this argument appeared somewhere else recently on MO, I think, but I forget where.