Hiroshi Okumura

Maebashi Gunma 371-0123 Japan
E-mail: hokmr@yandex.com

Резюме. We generalize a problem in W asan geometry involving an arbelos, and construct a self-similar circle pattern.

Ключови думи: arbelos; self-similar figure

1. Introduction

Let \(\alpha, \beta\) and \(\gamma\) be circles with diameters \(A O, B O\) and \(A B\) ，respectively，for a point \(O\) on the segment \(A B\) ．The configuration consisting of the three circles and the radical axis of \(\alpha\) and \(\beta\) are called an arbelos and the axis，respectively．The radii of \(\alpha\) and \(\beta\) are denoted by \(a\) and \(b\) ，respectively，and the reflection in the axis is denoted by \(\sigma\) ．In this note we generalize the following problem proposed by Satoh (佐藤幸吉定寄) in a Sangaku presented in Iwate in 1850 (Yasutomi, 1987) (see Fig. 1).

Problem 1. Show that the two circles touching \(\beta^{\sigma}\) externally, \(\gamma\) internally and the axis from the side opposite to \(B\), get radius \(a / 2\).

2. Generalization

Let \(\alpha(n)\) (resp. \(\beta(n)\) ) be the circle of radius (resp. \(n b\) ) touching the axis at \(O\) from the side opposite to \(A\) (resp. \(B\) ) for a non-negative real number \(n\). Let \(\alpha_{2}\) (resp. \(\beta_{2}\) ) be one of the two circles touching \(\beta(n)\) (resp. \(\alpha(n)\) ) externally, \(\gamma\) internally and the axis from the side opposite to \(B\) (resp. \(A\) ), where the circles \(\alpha_{2}\) and \(\beta_{2}\) lie on the same side of the segment \(A B\). Problem 1 is the case \(n=1\) in the next theorem (see Fig. 2).

Theorem 1. The circles \(\alpha_{2}\) and \(\beta_{2}\) have radii \(a /(n+1)\) and \(b /(n+1)\), respectively, and touch at a point on the axis.

Proof. We assume that \(D\) is the center of \(\alpha_{2}, F\) is the foot of perpendicular from \(D\) to \(A B\) and \(d=|D F|\). If \(r\) is the radius of \(\alpha_{2}\), then we get \((a+b-r)^{2}-((a-b)-r)^{2}=(n b+r)^{2}-(n b-r)^{2}=d^{2}\) from the two right triangles formed by \(D, F\) and the center of \(\gamma\), and \(D, F\) and the center of \(\beta(n)\). Solving the equations, we have \(r=a /(n+1)\) and \(d=2 \sqrt{n a b /(n+1)}\). Similarly \(\beta_{2}\) has radius \(b /(n+1)\). Since \(d\) is symmetric in \(a\) and \(b\), the distance from the center of \(\beta_{2}\) to \(A B\) also equals \(d\). Therefore the circle \(\alpha_{2}, \beta_{2}\) touch at a point on the axis.

Notice that if \(n=0\), then the circle \(\beta(n)\) is a point circle and the circles \(\alpha\) and \(\alpha_{2}\) coincide, also \(\beta\) and \(\beta_{2}\) coincide.

3. A self-similar circle pattern

In this section we construct a self-similar circle pattern by Theorem 1. We now consider the case \(n=1\) in the theorem as in Problem 1. W e denote the arbelos formed by \(\alpha, \beta\) and \(\gamma\) by (\(\alpha, \beta, \gamma\) ), and by, and denote the configuration consisting of \((\alpha, \beta, \gamma)\) and \((\alpha, \beta, \gamma)^{\sigma}\) by \(\mathcal{C}_{1}\), which is symmetric in the axis (see Fig. 3). Notice that the axis and the segment \(A B\) are not included in \(\mathcal{C}_{1}\). For a similar mapping \(\tau\), we call the figure \(\mathcal{C}_{1}^{\tau}\) a symmetric arbelos of radius \(\left|A^{\tau} B^{\tau}\right|\). Let us assume \(|A B|=1\).

Let \(\gamma_{2}\) be the smallest circle touching \(\alpha_{2}\) and \(\beta_{2}\) internally. Then the arbelos \(\left(\alpha_{2}, \beta_{2}, \gamma_{2}\right)\) is similar to \((\alpha, \beta, \gamma)\) by Theorem 1. We call the figure consisting of \(\left(\alpha_{2}, \beta_{2}, \gamma_{2}\right) \cup\left(\alpha_{2}, \beta_{2}, \gamma_{2}\right)^{\sigma}\) and its reflection in \(A B\) the two small copies of \(\mathcal{C}_{1}\), which consists of two symmetric arbeloi of radius \(1 / 2\). We define the two small copies of \(\mathcal{C}_{1}^{\tau}\) for a similar mapping \(\tau\) similarly. We denote the configuration consisting of \(\mathcal{C}_{1}\) and its two small copies by \(\mathcal{C}_{2}\) (see Fig. 4). If the figure \(\mathcal{C}_{k}\) is constructed, which consists of \(\mathcal{C}_{1}\), two symmetric arbeloi of radius \(1 / 2\), four symmetric arbeloi of radius \(1 / 2^{2}, \cdots, 2^{k-1}\) symmetric arbeloi of radius \(1 / 2^{k-1}\), then we add the two small copies of each of the \(2^{k-1}\) symmetric arbeloi of radius \(1 / 2^{k}\), and denote the resulting configuration by \(\mathcal{C}_{k+1}\). By this construction, we can get \(\mathcal{C}_{n}\) for any positive integer \(n\). Then \(\mathcal{C}_{0}=\bigcup_{i \gt 1} \mathcal{C}_{i}\) is a self-similar circle configuration (see Fig. 5).