Two pairs of Archimedean circles in the arbelos (Q2878862)

This is a joint review of [\textit{D. T. Oai}, ibid. 14, 201--202 (2014; Zbl 1306.51008)], [\textit{T. Q. Hung}, ibid., 249--251 (2014; Zbl 1306.51007)], [\textit{F. van Lamoen}, ibid., 253--254 (2014; Zbl 1306.51012)], [\textit{P. Yiu}, ibid., 255--260 (2014; Zbl 1306.51011)] and [\textit{H. Okumura}, ibid., 369--370 (2014; Zbl 1306.51009)].NEWLINENEWLINENEWLINENEWLINEThe Archimedean arbelos consists of three semicircles \(\Omega_A\), \(\Omega_B\), and \(\Omega\) having collinear diameters \(AP\), \(PB\), and \(AB\), respectively, and drawn on the same side of \(AB\), together with a line segment that is perpendicular to \(AB\) at \(P\) and that meets \(\Omega\) at \(D\). An interesting feature of this configuration is Archimedes' discovery that the incircles of the curvilinear triangles \(PDA\) and \(PDB\) are equal. These two circles, and any other circle of the same size as any of them, are referred to as Archimedean circles. Several Archimedean circles, usually coming in pairs, have already appeared in the recent literature.NEWLINENEWLINENEWLINENEWLINEIn the first paper under review, [Zbl 1306.51008] by \textit{D. T. Oai}, two more pairs of Archimedean circles are discovered. One of these pairs is obtained by drawing a tangent line that touches \(\Omega_A\) and \(\Omega_B\) and that meets \(\Omega\) at \(T_a\) and \(T_b\), then drawing the tangent lines \(L_a\) and \(L_b\) at \(T_a\) and \(T_b\) to \(\Omega\), then dropping perpendiculars \(DA'\) and \(DB'\) on \(L_a\) and \(L_b\). The circles having diameters \(DA'\) and \(DB'\) form a pair of Archimedean circles.NEWLINENEWLINENEWLINENEWLINEIn the second paper, [Zbl 1306.51007] by \textit{T. Q. Hung}, two more pairs are discovered. One of these pairs is obtained by drawing perpendiculars to \(AB\) from the centers of \(\Omega_A\) and \(\Omega_B\) that meet \(\Omega\) at \(E\) and \(F\), respectively, then letting \(AF\) and \(BE\) meet \(\Omega_A\) and \(\Omega_B\), respectively, at \(H\) and \(K\). The circles centered at \(H\) and \(K\) and that are tangent to \(PD\) form a pair of Archimedean circles.NEWLINENEWLINENEWLINENEWLINEIn the third paper, [Zbl 1306.51012], \textit{F. van Lamoen} examines the simple pair of Archimedean circles discovered in 2011 by Q. T. Bui, and later independently rediscovered by \textit{H. Okumura} in [Math. Gaz. 97, 512 (2013)]. This pair is obtained by taking \(E\) and \(F\) on \(\Omega\) such that \(EA=EP\) ad \(FB=FP\), and then letting \(A_1\) and \(A_2\) be the points where \(EA\) and \(EP\) meet \(\Omega_A\), and \(B_1\) and \(B_2\) be the points where \(FB\) and \(FP\) meet \(\Omega_B\). The Bui-Okumura twin circles are then the ones with diameters \(A_1A_2\) ad \(B_1B_2\). After proving that these circles are indeed Archimedean circles, he describes another approach to constructing them. The new approach gives rise to a distinguished point on \(PD\) that is realized as the intersection of two interesting loci.NEWLINENEWLINENEWLINENEWLINEIn the fourth paper, [Zbl 1306.51011], \textit{P. Yiu} describes explicit straightedge and compass constructions of three particular Archimedean circles. Completing the semicirclesNEWLINE\(\Omega_A\) and \(\Omega_B\) to circles \(\Omega_A^*\) and \(\Omega_B^*\), respectively, he considers Archimedean circles having a diameter \(UV\), with \(U\) lying on \(\Omega_A^*\) and \(V\) lying on \(\Omega_B^*\). In the first construction, \(UV\) is required to pass through \(P\); in the second, \(UV\) is required to be such that \(AU\) and \(BV\) meet on \(PD\); in the third, \(UV\) is required to be parallel to \(AB\). In each of the three cases, it is proved that \(UV\) is unique, except for a reflection about \(AB\).NEWLINENEWLINENEWLINENEWLINEThe last paper [Zbl 1306.51009] considers an Archimedean circle that was discovered by \textit{C. W. Dodge} et al. [Math. Mag. 72, No. 3, 202--213 (1999; Zbl 1015.51007)]. This is the circle that internally touches \(\Omega\) and that externally touches the two circles through \(P\) and having centers \(A\) and \(B\). Calling the line through the center of this circle and perpendicular to \(AB\) the Schoch line, the author of the paper under review discovers new Archimedean circles that are tangent to the Schoch line.