Pythagorean Theorem, Law of Sines and Law of Cosines: Alternative Proofs via shape Derivatives
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| Názov: | Pythagorean Theorem, Law of Sines and Law of Cosines: Alternative Proofs via shape Derivatives |
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| Autori: | Lorenzo Cavallina |
| Zdroj: | Mathematics Magazine. :1-6 |
| Publication Status: | Preprint |
| Informácie o vydavateľovi: | Informa UK Limited, 2025. |
| Rok vydania: | 2025 |
| Predmety: | Mathematics - History and Overview, History and Overview (math.HO), FOS: Mathematics |
| Popis: | 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. 6 pages, 4 figures |
| Druh dokumentu: | Article |
| Jazyk: | English |
| ISSN: | 1930-0980 0025-570X |
| DOI: | 10.1080/0025570x.2025.2535929 |
| DOI: | 10.48550/arxiv.2309.16691 |
| Prístupová URL adresa: | http://arxiv.org/abs/2309.16691 |
| Rights: | CC BY arXiv Non-Exclusive Distribution |
| Prístupové číslo: | edsair.doi.dedup.....b0ef938fae2e71e5bd3c2cf605863994 |
| Databáza: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstrakt: | 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.<br />6 pages, 4 figures |
|---|---|
| ISSN: | 19300980 0025570X |
| DOI: | 10.1080/0025570x.2025.2535929 |