The Archimedean Circles of Schoch and Woo

Transcription

1 Foum Geometicoum Volume 4 (2004) FRUM GEM ISSN The Achimedean Cicles of Schoch and Woo Hioshi kumua and Masayuki Watanabe Abstact. We genealize the Achimedean cicles in an abelos (shoemake s knife) given by Thomas Schoch and Pete Woo. 1. Intoduction Let thee semicicles α, β and γ fom an abelos, whee α and β touch extenally at the oigin. Moe specifically, α and β have adii a, b>0and centes (a, 0) and ( b, 0) espectively, and ae eected in the uppe half plane y 0. The y-axis divides the abelos into two cuvilinea tiangles. By a famous theoem of Achimedes, the inscibed cicles of these two cuvilinea tiangles ae conguent and have adii = ab a+b. See Figue 1. These ae called the twin cicles of Achimedes. Following [2], we call cicles conguent to these twin cicles Achimedean cicles. β(2b) U 2 α(2a) γ γ β α β α Figue 1 Figue 2 2 Fo a eal numbe n, denote by α(n) the semicicle in the uppe half-plane with cente (n, 0), touching α at. Similaly, let β(n) be the semicicle with cente ( n, 0), touching β at. In paticula, α(a) =α and β(b) =β. T. Schoch has found that (1) the distance fom the intesection of α(2a) and γ to the y-axis is 2, and (2) the cicle U 2 touching γ intenally and each of α(2a), β(2b) extenally is Achimedean. See Figue 2. P. Woo consideed the Schoch line L s though the cente of U 2 paallel to the y-axis, and showed that fo evey nonnegative eal numbe n, the cicle U n with cente on L s touching α(na) and β(nb) extenally is also Achimedean. See Figue 3. In this pape we give a genealization of Schoch s cicle U 2 and Woo s cicles U n. Publication Date: Mach 3, Communicating Edito: Paul Yiu.

2 28 H. kumua and M. Watanabe β(nb) L s β(2b) U n α(na) U 2 α(2a) Figue 3 2. A genealization of Schoch s cicle U 2 Let a and b be eal numbes. Conside the semicicles α(a ) and β(b ). Note that α(a ) touches α intenally o extenally accoding as a > 0 o a < 0; similaly fo β(b ) and β. We assume that the image of α(a ) lies on the ight side of the image of β(b ) when these semicicles ae inveted in a cicle with cente. Denote by C(a,b ) the cicle touching γ intenally and each of α(a ) and β(b ) at a point diffeent fom. Theoem 1. The cicle C(a,b ) has adius ab(a +b ) aa +bb +a b. α(a ) α(a ) β(b ) β(b ) b b a a a b b a Figue 4a Figue 4b Poof. Let x be the adius of the cicle touching γ intenally and also touching α(a ) and β(b ) each at a point diffeent fom. Thee ae two cases in which this cicle touches both α(a ) and β(b ) extenally (see Figue 4a) o one intenally and the othe extenaly (see Figue 4b). In any case, we have

3 The Achimedean Cicles of Schoch and Woo 29 (a b + b ) 2 +( x) 2 (b + x) 2 2(a b + b )( x) = (a (a b)) 2 +( x) 2 (a + x) 2 2(a, (a b))( x) by the law of cosines. Solving the equation, we obtain the adius given above. ab Note that the adius = a+b of the Achimedean cicles can be obtained by letting a = a and b,oa and b = b. Let P (a ) be the extenal cente of similitude of the cicles γ and α(a ) if a > 0, and the intenal one if a < 0, egading the two as complete cicles. Define P (b ) similaly. Theoem 2. The two centes of similitude P (a ) and P (b ) coincide if and only if a a + b b =1. (1) Poof. If the two centes of similitude coincide at the point (t, 0), then by similaity, a : t a = : t (a b) =b : t + b. Eliminating t, we obtain (1). The convese is obvious by the uniqueness of the figue. Fom Theoems 1 and 2, we obtain the following esult. Theoem 3. The cicle C(a,b ) is an Achimedean cicle if and only if P (a ) and P (b ) coincide. When both a and b ae positive, the two centes of similitude P (a ) and P (b ) coincide if and only if the thee semicicles α(a ), β(b ) and γ shae a common extenal tangent. Hence, in this case, the cicle C(a,b ) is Achimedean if and only if α(a ), β(b ) and γ have a common extenal tangent. Since α(2a) and β(2b) satisfy the condition of the theoem, thei extenal common tangent also touches γ. See Figue 5. In fact, it touches γ at its intesection with the y-axis, which is the midpoint of the tangent. The oiginal twin cicles of Achimedes ae obtained in the limiting case when the extenal common tangent touches γ at one of the intesections with the x-axis, in which case, one of α(a ) and β(b ) degeneates into the y-axis, and the emaining one coincides with the coesponding α o β of the abelos. Coollay 4. Let m and n be nonzeo eal numbes. Achimedean if and only if 1 m + 1 n =1. The cicle C(ma, nb) is

4 30 H. kumua and M. Watanabe Figue 5 3. Anothe chaacteizaton of Woo s cicles The cente of the Woo cicle U n is the point ( b a, 2 n + b + a ). (2) Denote by L the half line x =2, y 0. This intesects the cicle α(na) at the point ( 2, 2 ) (na ). (3) In what follows we conside β as the complete cicle with cente ( b, 0) passing though. Theoem 5. If T is a point on the line L, then the cicle touching the tangents of β though T with cente on the Schoch line L s is an Achimedean cicle. L s L T Figue 6

5 The Achimedean Cicles of Schoch and Woo 31 Poof. Let x be the adius of this cicle. By similaity (see Figue 6), b +2 : b =2 b a b + a : x. Fom this, x =. The set of Woo cicles is a pope subset of the set of cicles detemined in Theoem 5 above. The extenal cente of similitude of U n and β has y-coodinate 2a n +. When U n is the cicle touching the tangents of β though a point T on L, we shall say that it is detemined by T. The y-coodinate of the intesection of α and L is 2a a+b. Theefoe we obtain the following theoem (see Figue 7). Theoem 6. U 0 is detemined by the intesection of α and the line L : x =2. L s L T Figue 7 As stated in [2] as the popety of the cicle labeled as W 11, the extenal tangent of α and β also touches U 0 and the point of tangency at α coincides with the intesection of α and L. Woo s cicles ae chaacteized as the cicles detemined by the points on L with y-coodinates geate than o equal to 2a a+b. 4. Woo s cicles U n with n<0 Woo consideed the cicles U n fo nonnegative numbes n, with U 0 passing though. We can, howeve, constuct moe Achimedean cicles passing though points on the y-axis below using points on L lying below the intesection with α. The expession (2) suggests the existence of U n fo n<0. (4)

6 32 H. kumua and M. Watanabe In this section we show that it is possible to define such cicles using α(na) and β(nb) with negative n satisfying (4). Theoem 7. Fo n satisfying (4), the cicle with cente on the Schoch line touching α(na) and β(nb) intenally is an Achimedean cicle. Poof. Let x be the adius of the cicle with cente given by (2) and touching α(na) and β(nb) intenally, whee n satisfies (4). Since the centes of α(na) and β(nb) ae (na, 0) and ( nb, 0) espectively, we have ( b a b + a na ) 2 ( +4 2 n + ) =(x + na) 2, and ( ) b a 2 ( b + a + nb +4 2 n + ) =(x + nb) 2. Since both equations give the same solution x =, the poof is complete. 5. A genealization of U 0 We conclude this pape by adding an infinite set of Achimedean cicles passing though. Let x be the distance fom to the extenal tangents of α and β. By similaity, b a : b + a = x a : a. This implies x =2. Hence, the cicle with cente and adius 2 touches the tangents and the lines x = ±2. We denote this cicle by E. Since U 0 touches the extenal tangents and passes though, the cicles U 0, E and the tangent touch at the same point. We easily see fom (2) that the distance between the cente of U n and is 4n +1. Theefoe, U 2 also touches E extenally, and the smallest cicle touching U 2 and passing though, which is the Achimedean cicle W 27 in [2] found by Schoch, and U 2 touches E at the same point. All the Achimedean cicles pass though also touch E. In paticula, Bankoff s thid cicle [1] touches E at a point on the y-axis. Theoem 8. Let C 1 be a cicle with cente, passing though a point P on the x-axis, and C 2 a cicle with cente on the x-axis passing though. IfC 2 and the vetical line though P intesect, then the tangents of C 2 at the intesection also touches C 1. x 2 x 2 P P Figue 8a Figue 8b

7 The Achimedean Cicles of Schoch and Woo 33 Poof. Let d be the distance between and the intesection of the tangent of C 2 and the x-axis, and let x be the distance between the tangent and. We may assume 1 2 fo the adii 1 and 2 of the cicles C 1 and C 2.If 1 < 2, then 2 1 : 2 = 2 + d = x : d. See Figue 8a. If 1 > 2, then 1 2 : 2 = 2 : d 2 = x : d. See Figue 8b. In each case, x = 1. Let t n be the tangent of α(na) at its intesection with the line L. This is well defined if n b a+b. By Theoem 8, t n also touches E. This implies that the smallest cicle touching t n and passing though is an Achimedean cicle, which we denote by A(n). Similalay, anothe Achimedean cicle A (n) can be constucted, as the smallest cicle though touching the tangent t n of β(nb) at its intesection with the line L : x = 2. See Figue 9 fo A(2) and A (2). Bankoff s cicle is A ( ) 2 a = A ( ) 2 b, since it touches E at (0, 2). n the othe hand, U 0 = A(1) = A (1) by Theoem 6. L L s L β(2b) U 2 α(2a) γ β α E Figue 9 Theoem 9. Let m and n be positive numbes. The Achimedean cicles A(m) and A (n) coincide if and only if m and n satisfy 1 ma + 1 nb = 1 = 1 a + 1 b. (5) Poof. By (3) the equations of the tangents t m and t n ae (ma +(m 2)b)x +2 b(ma +(m 1)b)y =2mab, (nb +(n 2)a)x +2 a(nb +(n 1)a)y =2nab. These two tangents coincide if and only if (5) holds.

8 34 H. kumua and M. Watanabe The line t 2 has equation ax + b(2)y =2ab. (6) It clealy passes though ( 2b, 0), the point of tangency of γ and β (see Figue 9). Note that the point ( 2 ) a, 2 b(2) lies on E and the tangent of E is also expessed by (6). Hence, t 2 touches E at this point. The point also lies on β. This means that A(2) touches t 2 at the intesection of β and t 2. Similaly, A (2) touches t 2 at the intesection of α and t 2. The Achimedean cicles A(2) and A (2) intesect at the point ( b a b + a, ( a(a +2b)+ ) b(2)) on the Schoch line. Refeences [1] L. Bankoff, Ae the twin cicles of Achimedes eally twin?, Math. Mag., 47 (1974) [2] C. W. Dodge, T. Schoch, P. Y. Woo and P. Yiu, Those ubiquitous Achimedean cicles, Math. Mag., 72 (1999) Hioshi kumua: Depatment of Infomation Engineeing, Maebashi Institute of Technology, Kamisadoi Maebashi Gunma , Japan addess: okumua@maebashi-it.ac.jp Masayuki Watanabe: Depatment of Infomation Engineeing, Maebashi Institute of Technology, Kamisadoi Maebashi Gunma , Japan addess: watanabe@maebashi-it.ac.jp