On the Möbius geometry of Euclidean triangles (Q376530)

In this paper, the configuration consisting of a Euclidean triangle, its in-centre and its three ex-centres is studied from a Möbius geometric point of view. To this end, the triangle \((A,B,C)\) is considered as a quadrangle \((A,B,C,\infty)\) in the Möbius plane, i.e.\ the Euclidean plane extended by the point \(\infty\). Equivalently, the Möbius plane is the Riemann sphere (with the circles contained in it). Thus, it can also be seen as the ``boundary'' of hyperbolic \(3\)-space.NEWLINENEWLINEIt is shown that the quadrangle of in- and ex-centres of \((A,B,C)\) is the so-called ``conformal dual'' of \((A,B,C,\infty)\), and vice versa. Moreover, the conformal dual of an arbitrary quadrangle in Möbius geometry can be constructed in a purely Möbius geometric way. The two quadrangles under consideration can also be seen as ``ideal'' tetrahedra of hyperbolic \(3\)-space that are exchanged by an antipodal map. The symmetry group of the \(8\)-point configuration turns out to be the elementary abelian group of order \(8\), consisting of certain Möbius involutions.NEWLINENEWLINEThe various geometries occurring and their interrelations are described clearly and in detail. The results are inspired by another paper of the third author [Far East J. Math. Educ. 2, No. 1, 1--11 (2008; Zbl 1195.51023)].