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There are many examples on the Web of the use of Cavalieri's principle in determining areas and volumes of 2-D and 3-D geometrical figures. The Wikipedia link uses the principle as both a proof and ...

I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it? Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...