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arXiv · 2309.16691

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

Lorenzo Cavallina
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Abstract

We provide an alternative unified approach for proving the Pythagorean theorem (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 very simple: we translate a triangle along a specific direction and compute the resulting change in area. Equating the change in area to zero yields the statements of the three aforementioned theorems.

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Lorenzo Cavallina. 2023-08-05. Pythagorean Theorem, Law of Sines and Law of Cosines: alternative proofs via shape derivatives. https://arxiv.org/abs/2309.16691

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