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

TL;DR

This paper explores the Miquel-Steiner theorem across various geometric planes, revealing how the theorem's conditions adapt in elliptic, hyperbolic, Minkowski, and Galilean geometries, and identifying specific cases where the point's location can be precisely determined.

Contribution

It generalizes the Miquel-Steiner theorem to non-Euclidean and other geometric planes, detailing the modified conditions and special cases for different quadrilaterals.

Findings

In elliptic and hyperbolic planes, the theorem's intersection point becomes a radical center.

In Minkowski and Galilean planes, the circumcircles either touch at infinity or intersect at an anisotropic point.

For cyclic quadrilaterals, the Miquel-Steiner point's position can be explicitly calculated.

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.