On the sum of the angles between three vectors
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916959642189824 |
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| author | Pinelis, Iosif |
| author_facet | Pinelis, Iosif |
| contents | For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2π$. As an easy consequence of this, a proof of Euclid's theorem that the sum of the interior angles of any triangle is $π$ is provided. So, the main result of this note can be considered a generalization of Euclid's theorem. To a large extent, the consideration is reduced almost immediately to a choice for the sum of three related angles among the three integer multiples $0,2π,4π$ of $π$. The rest of the consideration concerns only various betweenness relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the sum of the angles between three vectors Pinelis, Iosif Metric Geometry 00A05, 01A20, 51A15, 51A20, 51A25, 51F99, 51G05, 51L99, 51M04, 51M05, 51M16, 51N10 For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2π$. As an easy consequence of this, a proof of Euclid's theorem that the sum of the interior angles of any triangle is $π$ is provided. So, the main result of this note can be considered a generalization of Euclid's theorem. To a large extent, the consideration is reduced almost immediately to a choice for the sum of three related angles among the three integer multiples $0,2π,4π$ of $π$. The rest of the consideration concerns only various betweenness relations. |
| title | On the sum of the angles between three vectors |
| topic | Metric Geometry 00A05, 01A20, 51A15, 51A20, 51A25, 51F99, 51G05, 51L99, 51M04, 51M05, 51M16, 51N10 |
| url | https://arxiv.org/abs/2508.04759 |