Dear Emerick:

The identity you cite is indeed often referred to as the Pythagorean Theorem. I would rather look at it as an abstract formulation of the Theorem. What the proof shows is that a certain algebraic identity holds. The identity is quite reminiscent of the Pythagorean Theorem and clearly maintains a semantic relationship to the latter.

In order for the proof of the "abstract" identity to be considered as a valid proof of the original theorem one has to establish that the vector norm has anything to do with the distance between two points (in an affine space, if you will). But in R² this would directly imply taking the Pythagorean Theorem for granted. A vicious circle.

Thank you,
Alex

Copyright © 1996-2009 Alexander Bogomolny

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