still don't think i understand the pythagorean theorem, all things considered
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it would take awhile to elaborate more fully on what i mean but here's a chunk of it: all the visual / geometric proofs take for granted that whatever "area" means it's something that's preserved under translations and rotations. it's unclear exactly what is needed for this
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Any two polygons of the same area can be subdivided into matching triangles like a tangram puzzle, and triangles can be compared for congruency by line segment length. It's maybe not obvious that area of this sort is well defined, but it is at least easy to compare for equality.
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yeah that's very clean and classical but it's famously known not to generalize to volume: