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Ubiquitous Archimedean Circles of the Collinear Arbelos

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Ubiquitous Archimedean Circles of the Collinear Arbelos KoG•16–2012 H. Okumura: Ubiquitous Archimedean Circles of the Collinear Arbelos Original scientific paper HIROSHI OKUMURA Accepted 5. 11. 2012. Ubiquitous Archimedean Circles of the Collinear Arbelos Ubiquitous Archimedean Circles of the Collinear Sveprisutne Arhimedove kruˇznice kolinearnog ar- Arbelos belosa ABSTRACT SAZETAKˇ We generalize the arbelos and its Archimedean circles, and Generaliziramo arbelos i njegove Arhimedove kruˇznice show the existence of the generalized Archimedean circles te pokazujemo postojanje generaliziranih Arhimedovih which cover the plane. kruˇznica koje pokrivaju ravninu. Key words: arbelos, collinear arbelos, ubiquitous Kljuˇcne rijeˇci: arbelos, kolinearni arbelos, sveprisutne Archimedean circles Arhimedove kruˇznice MSC 2000: 51M04, 51M15, 51N10 1 Introduction 2 The collinear arbelos For a point O on the segment AB in the plane, the area sur- In this section we generalize the arbelos and the twin cir- rounded by the three semicircles with diameters AO, BO cles of Archimedesto a generalizedarbelos. For two points and AB erected on the same side is called an arbelos. It P and Q in the plane, (PQ) denotes the circle with diameter has lots of unexpected but interesting properties (for an PQ. Let P and Q be point on the line AB, and let α = (AP), β γ extensive reference see [1]). The radical axis of the in- = (BQ) and = (AB). Let O be the point of intersection α β ner semicircles divides the arbelos into two curvilinear tri- of AB and the radical axis of the circles and and let angles with congruent incircles called the twin circles of u = |AB|, s = |AQ|/2 and t = |BP|/2. We use a rectangular Archimedes. Circles congruent to those circles are said coordinate system with origin O such that the points A, B have coordinate (a,0), (b,0) respectively with a − b = u. to be Archimedean. In this paper we generalize the arbe- The configuration (α,β,γ) is called a collinear arbelos if los and the Archimedean circles, and show the existence the four points lie in the order (i) B, Q, P, A or (ii) B, P, Q, of the generalized Archimedean circles covering the plane, A, or (iii) P, B, A, Q. In each of the cases the configura- which is a generalization of the ubiquitous Archimedean tion is explicitly denoted by (BQPA), (BPQA) and (PBAQ) circles of the arbelos in [4]. respectively. (BQPA) and (BPQA) are the generalized ar- The arbelos is generalized in several ways, the generalized belos of non-intersecting type and the generalized arbelos arbelos of intersecting type [7], the generalized arbelos of of intersecting type respectively. non-intersecting type [6] and the skewed arbelos [5], [8]. Let (p,0) and (q,0) be the coordinates of P and Q respec- For the generalized arbelos of intersecting type and non- tively. Since the point O lies on the radical axis of α and intersecting type, the twin circles of Archimedes are con- β, the powers of O with respect to α and β are equal, i.e., sidered in a general way as Archimedean circles in aliquot ap = bq holds. Hence there is a real number k < 0 such parts. But Archimedean circles are still not given except that b = ka and p = kq. Therefore we get them. In this paper we unify the two generalized arbeloi with one more additional generalized arbelos. ta + sb = tq + sp = 0. (1) 17 KoG•16–2012 H. Okumura: Ubiquitous Archimedean Circles of the Collinear Arbelos For points V and W on the line AB with x-coordinates v and w respectively, V ≤ W describes v ≤ w, and PW de- δβ γ notes the perpendicular to AB passing through W. The part δα (i) of the following lemma is proved in [3]. The proof of β (ii) is similar and is omitted (see Figure 1). α Lemma 1 The following circles have radii |AW||BV|/(2u) B Q O P A for pointsV andW on the line AB. γ δβ (i) The circles touching the circles internally, (AV) exter- δα γ nally and the line PW from the side opposite to the point B in the case B ≤ V ≤ A andB ≤ W ≤ A. β α (ii) The circles touching γ externally, (AV ) internally and PW from the side opposite to the point A in the case B P O Q A V ≤ B ≤ W ≤ A. P α O β δα δβ γ γ PAB O Q BVW A Figure 2: The circles δα and δβ. 3 A pair of Archimedean circles generated γ by a point We use the following lemma [4], which is easily proved by VWB A the properties of similar triangles. Figure 1: Circles of radii |AW||BV|/(2u). 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 RH δ For collinear arbeloi (BQPA) and (BPQA), α is the circle respectively such that SERF is a parallelogram. If T and in the region y > 0 touching the circles γ internally, α ex- U are points of intersection of RS with the lines parallel to ternally and the line P from the side opposite to B. For O GH passing through E and F respectively and g = |GS|, a collinear arbelos (PBAQ), δα is the circle in the region h = |HS|, then |ET| = |FU| = gh/(g + h). y > 0 touching γ externally, α internally and PO from the side opposite to A. The circle δβ is defined similarly (see Figure 2). Theorem 2 For a collinear arbelos (α,β,γ), let R be a point which does not lie on the line AB, and let E and F α β γ δ Theorem 1 For a collinear arbelos ( , , ), the circles α be points on the line AR and BR respectively such that EP δ and β are congruent with common radii st/(s +t). and FQ are parallel to BR and AR respectively. If the lines passing through E and F parallel to AB intersect the line Proof. If (α,β,γ) = (PBAQ), by (ii) of Lemma 1 and (1) OR at points T and U respectively, the circles (ET ) and the radius of δα is (FU) are Archimedean. |AO||BP| a(b − p) a(−ta/s +tq/s) st = = = . 2u 2(a − b) 2(a +ta/s) s +t Proof. Let EP and FQ intersect the line OR at points S Similarly the radius of δβ is equal to st/(s +t). The other and S0 respectively (see Figures 3, 4, 5). The triangles cases are proved similarly. RAO and S0QO are similar. Also the triangles RBO and We now call the circles δα and δβ the twin circles of SPO are similar. While |QO|/|AO| = |PO|/|BO| for O lies Archimedes of the collinear arbelos. Circles congruent on the radical axis of the circles α and β. Therefore the to the twin circles are called Archimedean circles of the ratios of the similarity of the two pairs of the similar tri- collinear arbelos. angles are the same. Hence |S0O|/|RO| = |SO|/|RO|, i.e., 18 KoG•16–2012 H. Okumura: Ubiquitous Archimedean Circles of the Collinear Arbelos S = S0. Let A0 and B0 be the pointsof intersection of the line 4 Ubiquitousness and parallelograms passing through S parallel to AB with the lines AR and BR respectively. Then |A0S| = |AQ| = 2s and |B0S| = |BP| = 2t. However the word “ubiquitous” is used in the title of the Therefore |ET | = |FU| = 2st/(s+t) by Lemma 2, i.e., the paper [2], their Archimedean circles do not cover the plane. circles (ET ) and (FU) are Archimedean. In this section, we show that our generalized Archimedean circles of the collinear arbelos cover the plane. Let (α,β,γ) be a collinear arbelos. If a point X does not lie on the line AB, let Y and Z be points such that the midpoint −→ −→ B’ S A’ of YZ is X, Y Z and AQ are parallel with the same direction, O and |Y Z| = 2rA. Let R be the point of intersection of the A B Q P lines AY and OZ. Using the point R, let us construct a par- E T allelogram SERF and a point T as in Figures 3, 4, 5. Then U (ET ) is an Archimedean circle with center X. F If a point X lies on the line AB, we choose a point X 0 lying 0 R inside of the Archimedean circle with center X, so that X does not lie on AB. If we usethe point X 0 instead of X, and Figure 3 construct the parallelogram SERF and the point T as in Figures 3, 4, 5 just as mentioned above, the Archimedean circle (ET ) with center X 0 contains X. Therefore there is an Archimedean circle containing the point X in any case, i.e., the Archimedean circles cover the plane. In this sense our Archimedean circles are Ubiquitous. However the five points A, B, P, Q and O are involved in the construction of the Archimedean circles, the three O B A α β γ P Q circles , and are not. Therefore it seems that the S Archimedean circles are not so closely related to the B’ A’ collinear arbelos. But we can show that for a point R, T E which does not lie on the line AB, the parallelogram SERF in Figures 3, 4, 5 are constructed by the circles α, β and γ F U (see Figure 6). Let the circle γ intersect the lines AR and BR at points I and J respectively, and let AJ intersect α at R a point K and BI intersect β at a point L.
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