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

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.

Figures (6)

• Figure 1: The light blue circle is the absolute conic in a hyperbolic plane.
• Figure 2: The red line is the Miquel-Steiner line of a quadrangle $\{A,B,C,D\}$ in the euclidean plane. The four blue circles form the only proper quadruple of tangent circles of this quadrangle.
• Figure 3: The light blue circle is the absolute conic in a hyperbolic plane.
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