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We provide an alternative unified approach for proving the Pythagoreantheorem (in dimension $2$ and higher), the law of sines and the law of cosines,based on the concept of shape derivative. The idea behind the proofs is verysimple: we translate a triangle along a specific direction and compute theresulting change in area. Equating the change in area to zero yields thestatements of the three aforementioned theorems.

Pythagorean Theorem, Law of Sines and Law of Cosines: alternative proofs via shape derivatives