Two Pairs of Archimedean Circles in the Arbelos

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Forum Geometricorum Volume 14 (2014) 201–202. FORUM GEOM ISSN 1534-1178 Two Pairs of Archimedean Circles in the Arbelos Dao Thanh Oai Abstract. We construct four circles congruent to the Archimedean twin circles in the arbelos. Consider an arbelos formed by semicircles (O1), (O2), and (O) of radii a, b, and a + b. The famous Archimedean twin circles associated in the arbelos have ab equal radii a+b (see [2, 3]). Let CD be the dividing line of the smaller semicircles, and extend their common tangent PQto intersect (O) at Ta and Tb. Theorem 1. Let A and B be the orthogonal projections of D on the tangents to (O) at Ta and Tb respectively. The circles with diameters DA and DB are congruent to the Archimedean twin circles. T A B D Ta P M Q Tb A O1 O C O2 B Figure 1 Proof. Let the tangents at Ta and Tb intersect at T . Since OT is the perpendicular bisector of TaTb, it intersects the semicircle (O) at the midpoint D of the arc TaTb (see [3, §5.2.1]). Since O1P , OM and O2Q are parallel, and O1P = OO2 = a, O2Q = O1O = b, a b a2 + b2 2ab OM = ·O P + ·O Q = =⇒ DM = OD−OM = . a + b 1 a + b 2 a + b a + b Publication Date: September 2, 2014. Communicating Editor: Floor van Lamoen. 202 T. O. Dao Now, ∠DTaT = ∠DTbTa = ∠DTaTb. Therefore, TaD bisects angle TTaTb. Similarly, TbD bisects angle TTbTa, and D is the incenter of triangle TTaTb.It follows that DA = DB = DM, and the circles with DA and DB are congruent to the Archimedean twin circles. Remark. The circle with DM as diameter is the Archimedean circle (A3) in [2] (or (W4) in [1]). Theorem 2. Let A1A2 and B1B2 be tangents to the smaller semicircles with A1, B1 on the line AB and A1A2 = a, B1B2 = b.IfH and K are the midpoints of the semicircles (O1) and (O2) respectively, and A = CH ∩ A1B2, B = CK ∩ B1A2, then the circles through C with centers A and B are congruent to the Archimedean twin circles. K H B2 A2 A B A1 A O1 C O O2 B B1 Figure 2 ◦ Proof. Clearly, ∠A CA1 = ∠HCO1 =45. Since B1B2 = O2B2 =√ b, ◦ ∠B2B1O2 =45,√ the lines CA and √B1B2 are parallel. Also, B1O2 = 2b. Similarly, A1O1 = 2a, and A1B1 =( 2 + 1)(a + b). Therefore, √ A1C ( 2+1)a ab CA = B1B2 · = b · √ = . A1B1 ( 2 + 1)(a + b) a + b ab Similarly, CB = a+b . Therefore, the circles through C with centers A and B are congruent to the Archimedean twin circles. References [1] C. W. Dodge, T. Schoch, P. Y. Woo and P. Yiu, Those ubiquitous Archimedean circles, Math. Mag., 72 (1999) 202–213. [2] F. M. van Lamoen, Online catalogue of Archimedean circles, http://home.kpn.nl/lamoen/wiskunde/Arbelos/Catalogue.htm [3] P. Yiu, Euclidean Geometry, Florida Atlantic University Lecture Notes, 1998, available at http://math.fau.edu/Yiu/Geometry.html Dao Thanh Oai: Cao Mai Doai, Quang Trung, Kien Xuong, Thai Binh, Viet Nam E-mail address: [email protected].

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