If you compare the area of the two squares shown in the figure, you can get a proof of the Pythagorean theorem.

For some reason this proof sticks in my mind and it is one of the first proofs in mathematics that I understood and remember. I believe I can give at least five proofs of this theorem of the top of my head.

In some sense this theorem is one of the cornerstones of modern mathematics. One can use it to build analytic geometry and the Cartesian coordinate system: it gives the definition of distance. One can also use it to define the notion of a metric in differential geometry: on small enough distances one has a distance function that satisfies the Pythagorean theorem. (The proper notion requires the tangent bundle to a manifold, but the meaning is clear intuitively.)

The reason I like this proof is that it suffices to translate the small triangles on the top to the configuration of the bottom. If one agrees that one can add and subtract areas (that area is an extensive concept as one would say in physics), then it is clear that the area of the smaller square on the top square is equal to the added area of the two smaller squares in the bottom. One does not even need the area law for triangles to make this one work and one can probably explain this to a very young kid/girl and he/she will still understand it: they can move the pieces on the square themselves.

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