Rational Steiner Porism

Transcription

1 Forum Geometricorum Volume 0. FORUM GEOM ISSN -8 Rational Steiner Porism Paul Yiu Abstract. We establish formulas relating the radii of neighboring circles in a Steiner chain, a chain of mutually tangent circles each tangent to two given ones, one in the interior of the other. From such we parametrize, for n =,,, all configurations of Steiner n-cycles of rational radii.. Introduction Given a circle OR and a circle Ir in its interior, we write the distance d between their centers in the form d = R r qrr. It is well known [, pp.8 00] that there is a closed chain of n mutually tangent circles each tangent internally to O and externally to I if and only if q = tan π n. In this case we have a Steiner n-cycle, and such an n-cycle can be constructed beginning with any circle tangent to both O and I. In this note we study the possibilities that the two given circles have rational radii and distance between their centers. Such is called a rational Steiner pair. Since cos π n = q +q, we must have cos π n rational. By a classic theorem in algebraic number theory, cos nπ is rational only for n =,, see, for example, [, p., Corollary.]. It follows that rational Steiner pairs exist only for q =,,, corresponding to n =,,. We shall give a parametrization of such pairs and proceed to show how to construct Steiner n-cycles consisting of circles of rational radii. Here are some examples of symmetric rational Steiner n-cycles for these values of n. Figure. Symmetric Steiner n-cycles for n =,, q n R,r,d Radii ρ,...,ρ n,,,,,,,,,,,0,,,,, Publication Date: November 0, 0. Communicating Editor: Li Zhou. The author thanks Li Zhou for comments and suggestions leading to improvements over an earlier version of this paper.

2 8 P. Yiu. Construction of Steiner chains In this section we consider two circles OR and Ir with centers at a distance d apart, without imposing any relation on R, r, d, nor rationality assumption. By a Steiner circle we mean one which is tangent to both O and I. We shall assume d 0 so that the circles O and I are not concentric. Clearly there are unique Steiner circles of radii ρ 0 := R r d and ρ := R r+d. For each ρ ρ 0, ρ, there are exactly two Steiner circles of radius ρ symmetric in the center line OI. The center of each is at distances R ρ from O and r + ρ from I. Proposition. If Aρ is a Steiner circle tangent to O at P and I at Q, then the line PQ contains T +, the internal center of similitude of O and I. A P Q O T + I Figure. Proof. Note that A divides OP internally in the ratio OA : AP = R ρ : ρ, so that ρ O + R ρp A =. R Similarly, the same point A divides IQ externally in the IA : AQ = r + ρ : ρ, so that ρ I + r + ρq A =. r Eliminating A from these two equations, and rearranging, we obtain rr ρp + Rr + ρq R + rρ = R I + r O. R + r This equation shows that a point on the line PQ is the same as a point on the line OI, which is the intersection of the lines PQ and OI. Note that the point on the line OI is independent of P. It is the internal center of similitude T + of the two circles, dividing O and I internally in the ratio OT + : T + I = R : r. Remark. The point T + can be constructed as the intersection of the line OI with the line joining the endpoints of a pair of oppositely parallel radii of the circles.

3 Rational Steiner porism Two Steiner circles are neighbors if they are tangent to each other externally. Proposition. If two neighboring Steiner circles are tangent to each other at T, then T lies on a circle with center T +. T P A T B T A A Q OT + I OT + I Figure Figure Proof. Applying the law of cosines to triangles POT + and AOI, we have R + Rd R+r T+ P R Rd R+r = R ρ + d r + ρ. R ρd From this, Similarly, It follows that T + P = R R + r d R + r r + ρ R ρ. T + Q = r R + r d R + r R ρ r + ρ. T + P T + Q = Rr R + r d R + r. If we put t = Rr R+r d independent on ρ, then the circle T R+r + t intersects the Steiner circle Aρ at a point T such that TT + is tangent to the Steiner circle see Figure. This leads to an easy construction of the neighbor of Aρ tangent at T see Figure : Extend TA to A such that TA = r. Construct the perpendicular bisector of IA to intersect the line AT at B. Then the circle BT is the Steiner circle tangent to Aρ at T.

4 0 P. Yiu. Radii of neighboring Steiner circles Henceforth we write R,r,d; ρ,..., ρ n for a rational Steiner pair R,r,d q and an n-cycles of Steiner circles with rational radii ρ,..., ρ n. To relate the radii of neighboring Steiner circles, we make use of the following results. Lemma. a ρ 0 ρ = qrr, b R ρ 0 R ρ = q + Rr. Proposition Bottema []. Given a triangle with sidelengths a, a, a, the distances d, d, d from the opposite vertices of these sides to a point in the plane of the triangle satisfy the relation d a + d + d a + d + d a + d + d d a + d + d a + d + d a + d + d d = 0. Proposition. Let Aρ be a Steiner circle between O and I. The radii of its two neighbors are the roots of the quadratic polynomial aσ + bσ + c, where a = q + Rr R rρ + Rrρ, b = q + Rrρq Rr R rρ, c = q + R r ρ. A ρ + σ R σ R ρ ρ + r B σ + r O d I Figure Proof. Let Bσ be a neighbor of Aρ. Apply Proposition to triangle IAB with sides ρ + σ, r + σ, r + ρ, and the point O whose distances from I, A, B are respectively d, R ρ, R σ. d R r + d R + rρ R r + d R + rσ R r + d R + rρ R ρ RR ρ R + ρσ R r + d R + rσ RR ρ R + ρσ R σ = 0.

5 Rational Steiner porism It is clear that the determinant is a quadratic polynomial in σ. With the relation, we eliminate d and obtain R r qrr R Rr qrr R + rρ R Rr qrr R + rσ R Rr qrr R + rρ R ρ RR ρ R + ρσ R Rr qrr R + rσ RR ρ R + ρσ R σ = 0. This determinant, apart from a factor, is aσ +bσ+c, with coefficients given by above. Lemma. b ac = q + R r ρ ρ ρρ ρ 0. Proof. b ac = q + R r ρ D, where D = q Rr R rρ q + Rr R rρ + Rrρ = Rr qrr + R rρ ρ = Rr ρ 0 ρ + ρ 0 + ρ ρ ρ = Rrρ ρρ ρ 0. Proposition. The radius ρ of a Steiner circle and those of its two neighbors are rational if and only if ρ = Rτ := τ ρ 0 + Rrρ τ + Rr for some rational number τ. Proof. The roots of the quadratic polynomial aσ + bσ + c are rational if and only if b ac is the square of a rational number. With a, b, c given in, this discriminant is given by Lemma. Writing Rrρ ρρ ρ 0 = τ ρ ρ 0 for a rational τ leads to the rational expression above. Theorem 8. For a Steiner circle with rational radius ρ = Rτ, the two neighbors have radii ρ + = Rτ + and ρ = Rτ where τ + = Rrτ ρ Rr + τρ 0 and τ = Rrτ + ρ Rr τρ 0. Proof. With ρ given by, we have i ρ ρ = τ ρ ρ 0 and ρ ρ τ +Rr 0 = Rrρ ρ 0 ii from τ +Rr, b = q + Rrρq Rr ρ 0 + ρ ρ, = q + Rrρ R + Rρ 0 + ρ + ρ 0 ρ ρ 0 + ρ τ ρ 0 + Rrρ τ + Rr = q + Rrρ Rr + ρ 0 τ + Rr + ρ Rr τ, + Rr

6 P. Yiu iii by Lemma, b ac = q + R r ρ ρ ρ 0 τ τ + Rr. The roots of the quadratic polynomial aσ + bσ + c are c b ε b ac = q + R r ρ q + Rrρ Rr+ρ 0 τ +Rr+ρ Rr ε q+r r ρρ ρ 0 τ τ +Rr τ +Rr q + Rrρ = = Rr+ρ 0 τ +Rr+ρ Rr ε Rrρ ρ 0 τ τ +Rr τ +Rr q + Rrτ ρ 0 + Rrρ Rr + ρ 0 τ + Rr + ρ Rr εrrρ ρ 0 τ, where ε = ±. On the other hand, Rrτ ερ R = R r τ ερ ρ 0 + RrRr + ετρ 0 ρ Rr + ετρ 0 R r τ ερ + RrRr + ετρ 0 = Rrτ ερ ρ 0 + Rr + ετρ 0 ρ Rrτ ερ + Rr + ετρ 0 = = Rr + ρ 0 ρ τ ρ 0 + Rrρ Rr + ρ 0 τ + Rr + ρ Rr εrrρ ρ 0 τ q + Rrτ ρ 0 + Rrρ Rr + ρ 0 τ + Rr + ρ Rr εrrρ ρ 0 τ. These are, according to above, the radii of the two neighbors.. Parametrizations A rational Steiner pair R, r, d is standard if R =. Proposition. The standard rational Steiner pairs are parametrized by R =, r = t qq + t, d = q + tq + + t q + tq + + t. Proof. Since R,r,d =,0, is a rational solution of d = r qr, every rational solution is of the form d = q + + tr for some rational number t. Direct substitution leads to q + tq + + tr t = 0. From this, the expressions of r and d follow. Remark. ρ 0 = t q++t and ρ = q q+t.

7 Rational Steiner porism Proposition 0. In a standard rational Steiner pair R,r,d q, a Steiner circle Aρ and its neighbors have rational radii if and only if ρ = Rτ = tq + q + t τ q + tt + q + tq + + tτ for a rational number τ. The radii of the two neighbors are Rτ ±, where τ + = q + q + tτ q + t + q + tτ and τ = q + q + tτ q + t q + tτ. Proof. The neighbors of Aρ have rational radii if and only if aσ +bσ+c with a, b, c given in has rational roots. Therefore, the two neighbors have rational radii if and only if Rrρ ρρ ρ 0 is the square of a rational number by Proposition, Theorem 8, and Proposition. Proposition. The iterations of τ + respectively τ have periods,, according as q =,, or. Proof. The iterations of τ + are as follows. ++tτ +t++tτ τ ++tτ +t++tτ +t τ τ ++tτ +t +tτ Figure. -cycle for q = ++tτ +t +tτ Figure. -cycle for q = ++tτ +t++tτ ++tτ +t++tτ +t τ τ ++tτ +t +tτ ++tτ +t +tτ Figure 8. -cycle for q = The iterations of τ simply reverse the orientations of these cycles.

8 P. Yiu Example. Rational Steiner -cycles with simple rational radii: t R, r, d τ Steiner cycle,,,, 0, 8, 0, 8, 8,,, 8,,, 8,, 8,, 8, 8,, 8, t R, r, d τ Steiner cycle,,, 0,,,,,,, 0, 8, 0 8,,,, 0, 0, 8,, 0,,, 0, 0, 8, 8 0, 0, 0, 0, A:,, ;, 0, 0 0 B:,, ; 8,, 8 0 C:,, ;,, D:,, ; 0,, Figure. Rational Steiner -cycles

9 Rational Steiner porism Example. Rational Steiner -cycles with simple rational radii: t R, r, d τ Steiner cycle,, 0,, 0,,,,, 0,, 8, 8, 0, 0,,,,,, 0,,, 0, 0, 0, 0,,, 0, 0, 88, 0, 0 8, 0, 0 0,, 0, 8 0, 0,, 0, 0, 0, A:,, ; 0, 0, 0, B:,, ; 0,, 0, C:,, ;, 0, 0, 0 8 D:,, ;,,, Figure 0. Rational Steiner -cycles

10 P. Yiu Example. Rational Steiner -cycles with simple rational radii: t R, r, d τ Steiner cycle, 8,,, 8, 0, 0 8, 8,, 8, 0 8, 0,,,,,,,, 8,,, 8, 8,, 8, 8, 8, 8,,, 0, 88,,, 8,,,,,,,,, 0, 8, 0, A:, 8, ;,, 8,, 8, 8 B:,, ; 8, 8,, 8, 8, C:,, ;,,,,, D:,, ;,,,,, 8 Figure. Rational Steiner -cycles

11 Rational Steiner porism. Inversion By inverting a Steiner n-cycle configuration R,r,d;ρ,...,ρ n in the circle O, we obtain a new n-cycle R,r,d ;ρ,...,ρ n in which r = R r d r, d = R d d r, ρ i = Rρ i, i =,...,n. ρ i R Regarding the circles in a Steiner configuration with I in the interior of O all positively oriented, we interpret circles with negative radii as those oppositely oriented to the circle O. The tables below show the rational Steiner n-cycles obtained by inverting those in Figures - in the circle O. Figure illustrates those obtained from the -cycles in Figure 0. Steiner -cycle Inversive images in O A, 0, 0 ;, 0,,, ;, 0, B,, ; 8,, 8,,; 8,, 8 C,, ;,,,, ; 8,, 0 D,, ; 0,,,, ; 0,, Steiner -cycle A,, ; 0 B,, ; 0, C, D, 8, 8 ;,,, 0, 0, 0,, 0,, ;, 0 8, 0, 0 Inversive images in O,, ; 0, 0, 0,,,; 0,, 0,,, ; 0, 0, 0, 0, 0, 0 ;,,, Steiner -cycle A, 8, ;,, 8,, 8, 8 Inversive images, 8, ;,, 8,, 8, B, 0, 0 ; 8, 8,, 8, 8, Inversive images,, ; 8, 8,, 8, 8 8, C,, ;,,,,, Inversive images,, ;,,, 8,, D,, ; 8,,,,, Inversive images,, ;,,,, 0, The rational Steiner pairs in Figures -A all have r > d. In the inversive images, I contains O in its interior. The new configuration is equivalent to a standard one with r <. For example, the Steiner pair in Figure A is equivalent to,,. The Steiner pairs in Figures -B all have r = d. The inversive image of I is a line at a distance r from O. In Figures -C and D, r < d. The images of O and I are disjoint circles. For the cycles in C, none of the Steiner circles contains O. Their images are all externally tangent to the images of O and I. On the other hand, for the cycles

12 8 P. Yiu in D, one of the Steiner circles contains O in its interior. Therefore, its inversive image contains those of O and I in its interior A:,, ; 0, 0, 0, B:,,; 0,, 0, C:,, ; 0, 0,, 0 0 D:,, ; 0 0,,, Figure. Rational Steiner -cycles by inversion. Relations among standard rational Steiner pairs We conclude this note with a brief explanation of the relations among Steiner pairs with different parameters. Denote by S q t the standard rational Steiner pair

13 Rational Steiner porism given in Proposition. By allowing t to take on negative values, we also include the disjoint pairs and those with I containing O. Proposition. Let O and I be the circles in S q t. a I is in the interior of O if and only if t > 0. b I contains O in its interior if and only if q + < t < q. c O and I are disjoint if and only if q < t < 0 or t < q +. Proof. a The circle I is contained in the interior of O if and only if d+r < R and d r > R. This means tq + + t > 0 and q + t > 0. Therefore, t > 0. b The circle I contains O in its interior if and only if d+r > R and d r < R. This means tq + + t < 0 and q + t < 0. Therefore, q + < t < q. c follows from a and b. Remarks. If q + < t < q, S q t is homothetic, by the homothety at I with ratio R r, to S qt, where t = q+q+t q++t > 0. For standard pairs S q t with t > 0, we may restrict to 0 < t < qq +. If t > qq +, S q t is the reflection of S qq+ q t in O. Proposition. The inversive image of S q t in the circle O is S q t. Proof. Let I r be the inversive image of I in O, with d = OI. r = R d + r R t = d r q tq + t, d = R d + r + R qq + t = d r q tq + t. From this it is clear that the inversive image of the pair S q t is S q t. Remarks. If t = 0, I reduces to a point on the circle O. If t = qq +, then d = 0. The circles O and I are concentric. If t = q or q +, the circle I degenerates into a line. This means that with t = q or q +, the circle I passes through O. It has radius q+. Therefore, the line in S q q is at a distance q + from O. If the circle I contains the center O, then the inversive image of I contains O. This means that q + < t < q, and q < t < q +. It follows that if 0 < t < q, the inversive images of the Steiner pair of circles are disjoint. References [] O. Bottema, On the distances of a point to the vertices of a triangle, Crux Math., 0 8. [] I. Niven, Irrational Numbers, MAA,. [] D. Pedoe, Geometry, A Comprehensive Course, Dover Reprint, 88. Paul Yiu: Department of Mathematical Sciences, Florida Atlantic University, Glades Road, Boca Raton, Florida -0, USA address: yiu@fau.edu