The Archimedean arbelos in three-dimensional space
The authors explore possible generalizations of the well-known Archimedes theorem on the arbelos [see i.e. \textit{L. Bankoff,} Math. Mag. 47, 214--218 (1974; Zbl 0291.50011) or \textit{H. Okumura} and \textit{M. Watanabe}, Forum Geom. 4, 27--34 (2004; Zbl 1069.51010)] to the real Euclidean \(3\)-space. Basis of the generalization is the so-called negative Soddy configuration which consists of \(4\) circles of the Euclidean plane: Let \(ABC\) be a triangle, then there exists a system of three mutual tangential circles \(\mathbf A\), \(\mathbf B\), \(\mathbf C\) centered at \(A\), \(B\), \(C\), respectively; the unique \textit{outer Soddy circle \(\mathbf Z\)} touches \(\mathbf A\), \(\mathbf B\), and \(\mathbf C\) and encloses \(\mathbf A\), \(\mathbf B\), and \(\mathbf C\). Above \(\mathbf A\), \(\mathbf B\), \(\mathbf C\), and \(\mathbf Z\) the authors erect hemispheres \(H_{\mathbf A}\), \(H_{\mathbf B}\), \(H_{\mathbf C}\), and \(H_{\mathbf Z}\), on the same side of the carrier plane of \(\mathbf A\), \(\mathbf B\), \(\mathbf C\), and \(\mathbf Z\). Denote by \(T_{\mathbf {AB}}\) the unique common tangent plane of \(H_{\mathbf A}\) and \(H_{\mathbf B}\), similarly \(T_{\mathbf {BC}}\) and \(T_{\mathbf {CA}}\) are defined. The \textit{Archimedean sphere} \(S_A\) touches \(H_{\mathbf A}\) externally, \(H_{\mathbf Z}\) internally and the two planes \(T_{\mathbf{AB}}\) and \(T_{\mathbf {AC}}\); the two further Archimedean spheres \(S_B\) and \(S_C\) are defined analogously. The authors give criteria for the following situations: 1. Two of the Archimedean spheres have equal radii. 2. Two of the Archimedean spheres coincide. 3. Three of the Archimedean spheres coincide. In order to prove the criteria the authors derive numerous formulas for the negative Soddy configuration. As a by-product of the investigations they get a proof of the theorem of L. Bankoff; cf. the first quotation of this review. Six figures accompany the considerations and calculations of the paper.
scientific article; zbMATH DE number 5320898