aa r X i v : . [ m a t h . HO ] J un A generalization of Pappus chain theorem
Hiroshi Okumura
Takahanadai Maebashi Gunma 371-0123, Japane-mail: [email protected]
Abstract.
We generalize Pappus chain theorem and give an analogue to thistheorem.
Keywords.
Pappus chain theorem
Mathematics Subject Classification (2010).
Introduction
Let α , β and γ be circles with diameters BC , CA and AB , respectively for apoint C on the segment AB . Pappus chain theorem says: if { α = δ , δ , δ , · · · } isa chain of circles whose members touch β and γ , the distance between the centerof the circle δ n and the line AB equals 2 nr n , where r n is the radius of δ n (seeFigure 1). In this article we give a simple generalization of this theorem and showthat if we consider a line passing through the centers of two circles in the chaininstead of AB , a similar theorem still holds.Figure 1.2. A generalization of Pappus chain theorem
Let { ε , ε , ε } = { α, β, γ } and { P , P , P } = { A, B, C } , where P P and P P are diameters of ε and ε , respectively. We consider the chain of circles C = {· · · , δ − , δ − , ε = δ , δ , δ , · · · } whose members touch the circles ε and ε .Let r n be the radius of δ n . Pappus chain theorem is obtained in the case i = 0 inthe following theorem (see Figure 2). Theorem 1. If D i is the center of the circle δ i ∈ C and H i ( n ) is the point ofintersection of the line P D i and the perpendicular to AB from D n , the followingrelation holds. (1) | D n H i ( n ) | = 2 | n − i | r n . Proof.
We invert the figure in the circle with center P orthogonal to δ n . Then δ n and P D i are fixed and ε and ε are inverted to the tangents of δ n perpendicularto AB . Let F be the foot of perpendicular from D n to AB . Since H i ( n ) is thecenter of the image of δ i , we have | H i ( n ) F | = 2 ir n , while | D n F | = 2 nr n . Hencewe get (1). (cid:3) A generalization of Pappus chain theorem
Figure 2: C = C α , i = 1, n = 33. An analogue to Pappus chain theorem
Let a and b be the radii of the circles α and β , respectively. We use a rect-angular coordinate system with origin C such that A and B have coordinates( − b,
0) and (2 a, ε = α , the chain is explicitly denoted by C α . The chains C β and C γ are defined similarly. Let c = a + b and let ( x n , y n ) bethe center coordinate of the circle δ n ∈ C . We have y n = 2 nr n by Pappus chaintheorem, and x n and r n are given in Table 1 [2, 3].Chain x n r n C α − b + bc ( b + c ) n a + bc abcn a + bc C β a − ca ( c + a ) n b + ca abcn b + ca C γ ab ( b − a ) n c − ab abcn c − ab Table 1: y n = 2 nr n Let l i,j ( i = j ) be the line passing through the centers of the circles δ i and δ j for δ i , δ j ∈ C . It is expressed by the equations bc − a ij ) x + a ( b + c )( i + j ) y − b (2 a ij − c ( b − c )) = 0 , ca − b ij ) x − b ( c + a )( i + j ) y + 2 a (2 b ij + c ( c − a )) = 0 , ab + c ij ) x + c ( a − b )( i + j ) y − ab ( a − b ) = 0(2)in the cases C = C α , C = C β , C = C γ , respectively.Let H i,j ( n ) be the point of intersection of the lines l i,j and x = x n with y -coordinate h i,j ( n ). Let d i,j ( n ) = h i,j ( n ) − y n , i.e., d i,j ( n ) is the signed distancebetween the center of δ n and H i,j ( n ). The following theorem is an analogue toPappus chain theorem (see Figure 3). It is also a generalization of [1]. Theorem 2. If i + j = 0 , then d i,j ( n ) = f i,j ( n ) r n holds, where (3) f i,j ( n ) = 2( n − i )( n − j ) i + j . Proof.
We consider the chain C α . By Table 1 and (2), we get h i,j ( n ) = 2( n + ij ) abc ( i + j )( n a + bc ) = 2 ( n + ij )( i + j ) r n . Therefore d i,j ( n ) = h i,j ( n ) − y n = 2 ( n + ij )( i + j ) r n − nr n = 2( n − i )( n − j ) i + j r n . The rest of the theorem can be proved in a similar way. (cid:3)
Figure 3: C = C β , { i, j } = { , } , n = 2 Corollary 1. If i = 0 in Theorem 2, the following statements hold. (i) If j = ± , d i,j ( n ) = ± n ( n ∓ r n . (ii) If j = ± , d i,j ( n ) = ± n ( n ∓ r n . Corollary 2. d i,j ( n ) − d i,j ( − n ) = − nr n for any integers i , j , n with i = ± j . References [1] A. Altintas, H. Okumura, A note on Pappus chain and a collinear theorem, Sangaku Journalof Mathematics, (2018) 11–12.[2] G. Lucca, Some identities arising from inversion of Pappus chains in an arbelos, ForumGeom., (2008) 171–174.[3] G. Lucca, Three Pappus chains inside the arbelos: some identities, Forum Geom.,7