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arXiv · 2603.24280

Quadri-Figures in Cayley-Klein Planes II: The Miquel-Steiner Theorem

Manfred Evers
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Abstract

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.

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Manfred Evers. 2026-03-25. Quadri-Figures in Cayley-Klein Planes II: The Miquel-Steiner Theorem. https://arxiv.org/abs/2603.24280

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