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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.24280 |
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| _version_ | 1866916042078420992 |
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| author | Evers, Manfred |
| author_facet | Evers, Manfred |
| contents | The Miquel-Steiner theorem for a quadrilateral in the Euclidean plane states that the circumcircles of the four component triangles intersect at a single point, which now is called the Miquel-Steiner point of the quadrilateral. In elliptic and in hyperbolic planes, the Miquel-Steiner theorem does not hold in this form. Instead, a weaker version applies: The circumcircles of the four component triangles of a quadrilateral have a common radical center, which we will also call the Miquel-Steiner point. The Miquel-Steiner theorem for Euclidean planes also needs to be modified for Minkowski and Galilean planes: Either the circumcircles of the four component triangles touch each other at a point on the line at infinity, or they intersect transversely at an anisotropic point. For specific quadrilaterals (such as cyclic quadrilaterals), the location of the Miquel-Steiner point can be determined more precisely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24280 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quadri-Figures in Cayley-Klein Planes II: The Miquel-Steiner Theorem Evers, Manfred Metric Geometry The Miquel-Steiner theorem for a quadrilateral in the Euclidean plane states that the circumcircles of the four component triangles intersect at a single point, which now is called the Miquel-Steiner point of the quadrilateral. In elliptic and in hyperbolic planes, the Miquel-Steiner theorem does not hold in this form. Instead, a weaker version applies: The circumcircles of the four component triangles of a quadrilateral have a common radical center, which we will also call the Miquel-Steiner point. The Miquel-Steiner theorem for Euclidean planes also needs to be modified for Minkowski and Galilean planes: Either the circumcircles of the four component triangles touch each other at a point on the line at infinity, or they intersect transversely at an anisotropic point. For specific quadrilaterals (such as cyclic quadrilaterals), the location of the Miquel-Steiner point can be determined more precisely. |
| title | Quadri-Figures in Cayley-Klein Planes II: The Miquel-Steiner Theorem |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2603.24280 |