We generalize the arbelos and its Archimedean circles, and show the existence of the generalized Archimedean circles which cover the plane.Generaliziramo arbelos i njegove Arhimedove kružnice te pokazujemo postojanje generaliziranih Arhimedovih kružnica koje pokrivaju ravninu

Hiroshi Okumura

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KoG•16–2012 H. Okumura: Ubiquitous Archimedean Circles of the Collinear ArbelosOriginal scientific paperAccepted 5. 11. 2012.HIROSHI OKUMURAUbiquitous Archimedean Circles ofthe Collinear ArbelosUbiquitous Archimedean Circles of the CollinearArbelosABSTRACTWe generalize the arbelos and its Archimedean circles, andshow the existence of the generalized Archimedean circleswhich cover the plane.Key words: arbelos, collinear arbelos, ubiquitousArchimedean circlesMSC 2000: 51M04, 51M15, 51N10Sveprisutne Arhimedove kružnice kolinearnog ar-belosaSAŽETAKGeneraliziramo arbelos i njegove Arhimedove kružnicete pokazujemo postojanje generaliziranih Arhimedovihkružnica koje pokrivaju ravninu.Ključne riječi: arbelos, kolinearni arbelos, sveprisutneArhimedove kružnice1 IntroductionFor a pointO on the segmentAB in the plane, the area sur-rounded by the three semicircles with diametersAO, BOandAB erected on the same side is called an arbelos. Ithas lots of unexpected but interesting properties (for anextensive reference see [1]). The radical axis of the in-ner semicircles divides the arbelos into two curvilinear tri-angles with congruent incircles called the twin circles ofArchimedes. Circles congruent to those circles are saidto be Archimedean. In this paper we generalize the arbe-los and the Archimedean circles, and show the existenceof the generalized Archimedean circles covering the plane,which is a generalization of the ubiquitous Archimedeancircles of the arbelos in [4].The arbelos is generalized in several ways, the generalizedarbelos of intersecting type [7], the generalized arbelos ofnon-intersecting type [6] and the skewed arbelos [5], [8].For the generalized arbelos of intersecting type and non-intersecting type, the twin circles of Archimedes are con-sidered in a general way as Archimedean circles in aliquotparts. But Archimedean circles are still not given exceptthem. In this paper we unify the two generalized arbeloiwith one more additional generalized arbelos.2 The collinear arbelosIn this section we generalize the arbelos and the twin cir-cles of Archimedes to a generalized arbelos. For two pointsP andQ in the plane,(PQ) denotes the circle with diameterPQ. LetP andQ be point on the lineAB, and letα = (AP),β = (BQ) andγ = (AB). Let O be the point of intersectionof AB and the radical axis of the circlesα andβ and letu = |AB|, s = |AQ|/2 andt = |BP|/2. We use a rectangularcoordinate system with originO such that the pointsA, Bhave coordinate(a,0), (b,0) respectively witha− b = u.The configuration(α,β,γ) is called a collinear arbelos ifthe four points lie in the order (i)B, Q, P, A or (ii) B, P, Q,A, or (iii) P, B, A, Q. In each of the cases the configura-tion is explicitly denoted by(BQPA), (BPQA) and(PBAQ)respectively.(BQPA) and(BPQA) are the generalized ar-belos of non-intersecting type and the generalized arbelosof intersecting type respectively.Let (p,0) and(q,0) be the coordinates ofP andQ respec-tively. Since the pointO lies on the radical axis ofα andβ, the powers ofO with respect toα andβ are equal, i.e.,ap = bq holds. Hence there is a real numberk < 0 suchthatb = ka andp = kq. Therefore we getta + sb = tq + sp = 0. (1)17KoG•16–2012 H. Okumura: Ubiquitous Archimedean Circles of the Collinear ArbelosFor pointsV andW on the lineAB with x-coordinatesvandw respectively,V ≤ W describesv ≤ w, andPW de-notes the perpendicular toAB passing throughW . The part(i) of the following lemma is proved in [3]. The proof of(ii) is similar and is omitted (see Figure 1).Lemma 1 The following circles have radii |AW ||BV |/(2u)for points V and W on the line AB.(i) The circles touching the circles γ internally, (AV ) exter-nally and the line PW from the side opposite to the point Bin the case B ≤V ≤ A and B ≤W ≤ A.(ii) The circles touching γ externally, (AV ) internally andPW from the side opposite to the point A in the caseV ≤ B ≤W ≤ A.γB VV W AAγBWFigure 1: Circles of radii |AW ||BV |/(2u).For collinear arbeloi(BQPA) and(BPQA), δα is the circlein the regiony > 0 touching the circlesγ internally,α ex-ternally and the linePO from the side opposite toB. Fora collinear arbelos(PBAQ), δα is the circle in the regiony > 0 touchingγ externally,α internally andPO from theside opposite toA. The circleδβ is defined similarly (seeFigure 2).Theorem 1 For a collinear arbelos (α,β,γ), the circles δαand δβ are congruent with common radii st/(s+ t).Proof. If (α,β,γ) = (PBAQ), by (ii) of Lemma 1 and (1)the radius ofδα is|AO||BP|2u=a(b− p)2(a−b)=a(−ta/s+ tq/s)2(a + ta/s)=sts+ t.Similarly the radius ofδβ is equal tost/(s+ t). The othercases are proved similarly. We now call the circlesδα and δβ the twin circles ofArchimedes of the collinear arbelos. Circles congruentto the twin circles are called Archimedean circles of thecollinear arbelos.APOQBγδαδβαβAOBγαβP Qδαδβδα δβOP AB QγαβPOFigure 2: The circles δα and δβ.3 A pair of Archimedean circles generatedby a pointWe use the following lemma [4], which is easily proved bythe properties of similar triangles.Lemma 2 For a triangle RGH with a point S on the seg-ment GH, let E and F be points on the lines RG and RHrespectively such that SERF is a parallelogram. If T andU are points of intersection of RS with the lines parallel toGH passing through E and F respectively and g = |GS|,h = |HS|, then |ET | = |FU |= gh/(g + h).Theorem 2 For a collinear arbelos (α,β,γ), let R be apoint which does not lie on the line AB, and let E and Fbe points on the line AR and BR respectively such that EPand FQ are parallel to BR and AR respectively. If the linespassing through E and F parallel to AB intersect the lineOR at points T and U respectively, the circles (ET ) and(FU) are Archimedean.Proof. Let EP andFQ intersect the lineOR at pointsSand S′ respectively (see Figures 3, 4, 5). The trianglesRAO andS′QO are similar. Also the trianglesRBO andSPO are similar. While|QO|/|AO|= |PO|/|BO| for O lieson the radical axis of the circlesα andβ. Therefore theratios of the similarity of the two pairs of the similar tri-angles are the same. Hence|S′O|/|RO| = |SO|/|RO|, i.e.,18KoG•16–2012 H. Okumura: Ubiquitous Archimedean Circles of the Collinear ArbelosS = S′. LetA′ andB′ be the points of intersection of the linepassing throughS parallel toAB with the linesAR andBRrespectively. Then|A′S|= |AQ|= 2s and|B′S|= |BP|= 2t.Therefore|ET |= |FU |= 2st/(s+ t) by Lemma 2, i.e., thecircles(ET ) and(FU) are Archimedean. AOBRESFA’B’PQTUFigure 3ABOREFP QSUTA’B’Figure 4EFROSTUA’ B’ABP QFigure 54 Ubiquitousness and parallelogramsHowever the word “ubiquitous” is used in the title of thepaper [2], their Archimedean circles do not cover the plane.In this section, we show that our generalized Archimedeancircles of the collinear arbelos cover the plane.Let (α,β,γ) be a collinear arbelos. If a pointX does not lieon the lineAB, letY andZ be points such that the midpointof YZ is X ,−→Y Z and−→AQ are parallel with the same direction,and|Y Z| = 2rA . Let R be the point of intersection of thelinesAY andOZ. Using the pointR, let us construct a par-allelogramSERF and a pointT as in Figures 3, 4, 5. Then(ET ) is an Archimedean circle with centerX .If a pointX lies on the lineAB, we choose a pointX ′ lyinginside of the Archimedean circle with centerX , so thatX ′does not lie onAB. If we use the pointX ′ instead ofX , andconstruct the parallelogramSERF and the pointT as inFigures 3, 4, 5 just as mentioned above, the Archimedeancircle (ET ) with centerX ′ containsX . Therefore there isan Archimedean circle containing the pointX in any case,i.e., the Archimedean circles cover the plane. In this senseour Archimedean circles are Ubiquitous.However the five pointsA, B, P, Q and O are involvedin the construction of the Archimedean circles, the threecircles α, β and γ are not. Therefore it seems that theArchimedean circles are not so closely related to thecollinear arbelos. But we can show that for a pointR,which does not lie on the lineAB, the parallelogramSERFin Figures 3, 4, 5 are constructed by the circlesα, β andγ(see Figure 6). Let the circleγ intersect the linesAR andBR at pointsI andJ respectively, and letAJ intersectα ata pointK andBI intersectβ at a pointL. ThenKP andJB are parallel, alsoLQ andIA are parallel. Therefore ifE, F , S are the points of intersection of the linesAR andKP, BR andLQ, KP andLQ respectively, thenSERF is aparallelogram.AQRIJEFLKSPBFigure 619KoG•16–2012 H. Okumura: Ubiquitous Archimedean Circles of the Collinear ArbelosReferences[1] H. P. BOAS, Reflections on the Arbelos,Amer. Math.Monthly 113 (2006), 236–249.[2] C. W. DODGE, T. SCHOCH, P. Y. WOO, P. YIU,Those ubiquitous Archimedean circles,Math. Mag.72 (1999), 202–213.[3] R. NAKAJIMA , H. OKUMURA , Archimedean circlesinduced by skewed arbeloi,Journal for Geometryand Graphics 16 (2012), 13–17.[4] H. OKUMURA , Ubiquitous Archimedean circles,Mathematics and Informatics 55 (2012), 308–311.[5] H. OKUMURA , More on twin circles on the skewedarbelos,Forum Geom. 11 (2011), 139–144.[6] H. OKUMURA , M. WATANABE , Generalized arbe-los in aliquot parts: non-Intersecting case,Journalfor Geometry and Graphics 13 (2009), 41–57.[7] H. OKUMURA , M. WATANABE , Generalized Arbe-los in aliquot parts: intersecting case,Journal for Ge-ometry and Graphics 12 (2008), 53–62.[8] H. OKUMURA , M. WATANABE , The twin circles ofArchimedes in a skewed arbelos,Forum Geom. 4(2004), 229–251.Hiroshi Okumurae-mail: hiroshiokmr@gmail.com251 Moo 15 Ban Kesorn Tambol SilaAmphur Muang Khonkaen 40000, Thailand20

Sveprisutne Arhimedove kružnice kolinearnog arbelosa

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