Exparabolas of a Triangle
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914436506189824 |
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| author | Lukarevski, Martin Schröcker, Hans-Peter |
| author_facet | Lukarevski, Martin Schröcker, Hans-Peter |
| contents | Among a triangle's exparabolas (parabolas escribed to the triangle), three are distinguished by having locally maximal parameter. They are determined by a simple cubic equation and characterized by having axes that contain the triangle's centroid. More generally, there are three (not necessarily real) exparabolas with axes through a given point $X$. Their focal points determine another triangle which we call the $X$-focal triangle. It shares the circumcircle with the original triangle and its orthocenter is $X$. The sequence of iterated focal triangles with respect to the centroids splits into an even and an odd sub-sequence that both converge to equilateral triangles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exparabolas of a Triangle Lukarevski, Martin Schröcker, Hans-Peter Metric Geometry 51M04 51N20 Among a triangle's exparabolas (parabolas escribed to the triangle), three are distinguished by having locally maximal parameter. They are determined by a simple cubic equation and characterized by having axes that contain the triangle's centroid. More generally, there are three (not necessarily real) exparabolas with axes through a given point $X$. Their focal points determine another triangle which we call the $X$-focal triangle. It shares the circumcircle with the original triangle and its orthocenter is $X$. The sequence of iterated focal triangles with respect to the centroids splits into an even and an odd sub-sequence that both converge to equilateral triangles. |
| title | Exparabolas of a Triangle |
| topic | Metric Geometry 51M04 51N20 |
| url | https://arxiv.org/abs/2511.22347 |