Diophantine Steiner Triples and Pythagorean-Type Triangles

Transcription

1 Forum Geometricorum Volume 10 (2010) FORUM GEOM ISSN Diophntine Steiner Triples nd Pythgoren-Type Tringles ojn Hvl bstrct. We present connection between Diophntine Steiner triples (integer triples relted to configurtions of two circles, the lrger contining the smller, in which the Steiner chin closes) nd integer-sided tringles with n ngle of 60, 90 or 120. We introduce n explicit formul nd provide geometricl interprettion. 1. Introduction In [3] we described ll integer triples (R, r, d), R>r+ d, for which configurtion of two circles of rdii R nd r with the centers d prt possesses closed Steiner chin. This mens tht there exists cyclic sequence of n circles L 1,...,L n ech tngent to the two circles of rdii R nd r, nd to its two neighbors in the sequence. Such triples re clled Diophntine Steiner (DS) triples. For obvious resons the considertion cn be limited to primitive DS triples, i.e., DS triples with gcd(r, r, d) =1. We lso proved in [3] tht the only possible length of Steiner chin in DS triple is 3, 4 or 6. Therefore, the set of primitive DS triples cn be divided into three disjoint sets DS n for n =3, 4, 6. The elements of these sets re solutions of the following Diophntine equtions: The sequence of R in DS 4 is n reltion 3 R 2 14Rr + r 2 d 2 =0, 4 R 2 6Rr + r 2 d 2 =0, 6 3R 2 10Rr +3r 2 3d 2 =0. 6, 15, 20, 28,... In the ENYLOPEDI OF INTEGER SEQUENES (EIS) [6], this is the sequence of semi-perimeters of Pythgoren tringles. This suggested to us tht DS 4 might be closely connected with the Pythgoren tringles. It turns out tht in the sme mnner the sets DS 3 nd DS 6 re connected with integer sided tringles hving n ngle of 120 or of 60, respectively. Such tringles were considered in Publiction Dte: September 13, ommunicting Editor: Pul Yiu.

2 94. Hvl ppers [1, 4, 5]. Together with Pythgoren tringles, these form set of tringles tht we will cll Pythgoren-type tringles. It is surprising tht bijective correspondences between three pirs of triples sets re given by the sme formul (Theorem 1 below). It is the purpose of this pper to present this formul, provide geometricl interprettion nd derive some further curiosities. 2. ijective correspondence between the sets Q ϕ nd DS n The sides of Pythgoren-type tringles form three sets of triples, which we denote by Q 60, Q 90 nd Q 120 respectively. The set Q ϕ contins ll primitive integer triples (, b, c) such tht tringle with the sides, b, c contins the ngle ϕ degrees opposite to side c. We lso require b>. (This excludes the triple (1, 1, 1) from Q 60 nd voids dupliction of triples with the roles of nd b interchnged.) It is lso convenient to slightly modify the sets Q 60 nd Q 120 to sets Q 60 nd Q 120 s follows: triple (, b, c) with three odd numbers, b, c is replced with triple (2, 2b, 2c). Other triples remin unchnged. Modifiction in Q 90 is not necessry, since primitive Pythgoren triples lwys include exctly one even number. Theorem 1. The correspondences DS 4 Q 90, DS 3 Q 120 nd DS 6 Q 60 given by (R, r, d) ( 1 2 (R + r d), 1 2 (R + r + d),r r) (1) (, b, c) ( 1 2 ( + b + c), 1 2 ( + b c),b ) (2) re bijective nd inverse to ech other. Proof. It is strightforwrd tht the bove mps re mutully inverse nd tht they mp the solution (R, r, d) of the eqution R 2 6Rr +r 2 d 2 =0into the solution (, b, c) of the eqution 2 +b 2 c 2 =0, nd vice vers. The sme could be proved for the pir R 2 14Rr + r 2 d 2 =0nd 2 + b 2 + b c 2 =0, s well s for the pir 3R 2 10Rr +3r 2 3d 2 =0nd 2 + b 2 b c 2 =0. Using stndrd rguments, we lso prove tht the given primitive triple of DS 4 corresponds to the primitive triple of Q 90, nd vice vers. In the other two cses, considertion is similr but with slight difference: the triples from Q 60 nd Q 120 cn hve three odd components; therefore, the multipliction by 2 ws needed. Now we prove tht triples (R, r, d) from DS 3 nd DS 6 with n even d correspond to the modified triples of Q 60 nd Q 120 of the form (2, 2b, 2c),,b,cbeing odd; nd triples with odd d correspond to the untouched triples of Q 60 nd Q 120. In ech cse, the primitiveness of the triples from Q 60 nd Q 120 implies the primitiveness of those from DS 3 nd DS 6, nd vice vers. Remrk. Without restriction to integer vlues, these correspondences extend to the configurtions (R, r, d) with Steiner chins of length n = 3, 4, 6 nd tringles contining n ngle 180 n.

3 Diophntine Steiner triples nd Pythgoren-type tringles Geometricl interprettion We present geometricl interprettion of the reltions (1) nd (2). Let (R, r, d) be DS triple from DS n, n {3, 4, 6}. eginning with two points S 1, S 2 t distnce d prt, we construct two circles S 1 (R) nd S 2 (r). Let the line S 1 S 2 intersect the circle S 1 (R) t the points U, V nd S 2 (r) t W nd Z. On opposite sides of S 1 S 2, construct two similr isosceles tringles VUI c nd WZI on the segments UV nd WZ, with ngle 180 n between the legs. omplete the tringle with I s incenter I. Then I c is the excenter on the side c long the line S 1 S 2. This is the corresponding Pythgoren-type tringle (see Figure 1 for the cse of n =4). To prove this, it is enough to show tht the sides of tringle re = 1 2 (R + r d), b= 1 2 (R + r + d), c= R r, i.e. the sides given in (1). I c V S 1 S 2 W Z U I Figure 1. This construction yields tringle with given incircle, -excircle nd their touching points with side c. To clculte sides, b nd c, we mke use of the following formuls, where r i, r c, nd d re the inrdius, -exrdius, nd the distnce between the touching points: ( 4ri ) = 1 2 r c + d 2 rc+r i r c r i d, ( 4ri b = 1 2 r c + d 2 rc+r i r c r i + d c = 4r i r c + d 2. ),

4 96. Hvl Now let us consider different n {3, 4, 6}. In the cse n =3, ccording to the construction, r i = r 3, r c = R 3 nd 4r i r c + d 2 =12Rr + d 2. Since triples from DS 3 stisfy R 2 14Rr + r 2 d 2 =0,wehve12Rr + d 2 =(R r) 2. Hence, 4r i r c + d 2 = R r. For n =4nd n =6, we hve different r i nd r c nd pply different Diophntine equtions, but end up with the sme vlue of the squre root. pplying ll these to the formuls bove, we get the desired sides, b, c. 4. The reltion between sets DS 3 nd DS 6 In [3] we found n injective (but not surjective) mp from DS 3 to DS 6. In this section, we will explin the bckground nd provide geometricl interprettion of this reltion. In 2, we hve the bijections DS 3 Q 120 Q 120 nd DS 6 Q 60 Q 60. esides, it is cler from Figure 2 tht the mp (, b, c) (, + b, c) represent n injective mp from Q 120 to Q 60. D b c Figure 2. The sme is true for the mp (, b, c) (b, + b, c). We therefore hve two mps Q 120 Q 60, the union of their disjoint imges being the whole Q 60. Therefore, the sequence of mps DS 3 Q 120 Q 120 Q 60 Q 60 DS 6 defines two mps DS 3 DS 6. Following step by step, we cn esily find both explicit formuls: g 1 (R, r, d) = k ( 1 4 (5R + r d), 1 4 (R +5r d), 1 2 (R + r + d)) g 2 (R, r, d) = k ( 1 4 (5R + r + d), 1 4 (R +5r + d), 1 2 (R + r d)) with the pproprite fctor k {2, 1, 1 2 }. The correspondence noticed in [3] is, in fct, just g 2 with the chosen mximl possible fctor k =2(in multiplying by lrger fctor, we only lose primitiveness). The existence of two mps g 1 nd g 2 whose imges cover DS 6 explins why the imge of g 2 lone covered only one hlf of DS 6. Now we give geometric interprettion of these mps. Let us strt with triple (R, r, d) DS 3 nd construct the ssocited tringle from Q 120. ccording to (1), the sides nd b of this tringle re = R+r d 2 nd b = R+r+d 2. Hence, g 1 (R, r, d) = k (R + 2,r + 2,b). To get the first possible configurtion of two circles with the closed Steiner triple of the length n =6, we drw circles with centers nd with the rdii R = R+ 2 nd r = r+ 2 (see Figure 3). Similrly, drwing the circles with centers nd with the rdii R = R+ b 2 nd r = r+ b 2,

5 Diophntine Steiner triples nd Pythgoren-type tringles 97 Figure 3. we get the second possibility, rising from g 2. To obtin triples in DS 6, we must consider the effect of k: i.e., it is possible tht the elements (R,r,d ) need to be multiplied or divided by 2. References [1]. urn, Tringles with 60 ngle nd sides of integer length, Mth. Gzette, 87 (2003) Number Mrch, [2] H. S. M. oxeter nd S. L. Greitzer, Geometry revisited, Mth. ssoc. mer., Wshington D, [3]. Hvl, Diophntine Steiner triples, to pper in Mth. Gzette. [4] E. Red, On integer-sided tringles contining ngles of 120 or 60, Mth. Gzette, 90 (2006) Number July, [5] K. Selkirk, Integer-sided tringles with n ngle of 120, Mth. Gzette, 67 (1983) Number Dec, [6] N.J.. Slone, The On-line Encyclopedi of Integer Sequences, vilble t njs/sequences/. ojn Hvl: Deprtment of Mthemtics nd omputer Science, FNM, University of Mribor, Korošk cest 160, 2000 Mribor, Sloveni E-mil ddress: bojn.hvl@uni-mb.si