The Perimeter Winternitz Theorem in a Triangle
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908912015376384 |
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| author | Berele, Allan Catoiu, Stefan |
| author_facet | Berele, Allan Catoiu, Stefan |
| contents | A variable line through the centroid G of a triangle divides the triangle into two parts each of whose lengths as a fraction of the perimeter fills a closed interval [m,1-m], with m between 0 and 1/2. We show that the range of m taken over all triangles is the interval (3/10,4/9], with 3/10 approached by scales of the triangles approaching the 5-4-1 triangle and their mid-size medians, and 4/9 attained by the equilateral triangles and the lines through G parallel to the sides. This result is the perimeter version of the classical Winternitz theorem for a triangle, asserting that, in the case of area-ratio instead of perimeter-ratio, m=4/9, and this is attained by all triangles and their lines through G and parallel to the sides. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23653 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Perimeter Winternitz Theorem in a Triangle Berele, Allan Catoiu, Stefan Metric Geometry Primary: 52A10, 52A38. Secondary: 51M04, 51M25, 51N20 A variable line through the centroid G of a triangle divides the triangle into two parts each of whose lengths as a fraction of the perimeter fills a closed interval [m,1-m], with m between 0 and 1/2. We show that the range of m taken over all triangles is the interval (3/10,4/9], with 3/10 approached by scales of the triangles approaching the 5-4-1 triangle and their mid-size medians, and 4/9 attained by the equilateral triangles and the lines through G parallel to the sides. This result is the perimeter version of the classical Winternitz theorem for a triangle, asserting that, in the case of area-ratio instead of perimeter-ratio, m=4/9, and this is attained by all triangles and their lines through G and parallel to the sides. |
| title | The Perimeter Winternitz Theorem in a Triangle |
| topic | Metric Geometry Primary: 52A10, 52A38. Secondary: 51M04, 51M25, 51N20 |
| url | https://arxiv.org/abs/2603.23653 |