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Amicable Triangles and Perfect Circles

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Amicable Triangles and Perfect Circles Journal for Geometry and Graphics Volume 17 (2013), No. 1, 53{67. Amicable Triangles and Perfect Circles Michael Sejfried ul. Zyzna_ 11F, PL-42200 Cz¸estochowa, Poland email: [email protected] Abstract. This is a contribution to triangle geometry. Two amicable triangles are inscribed in any circle which is related to a reference triangle ABC. Amicable triangles give rise to some family of circles { so-called perfect circles. In this way it is possible to generalize geometrical objects like the Soddy and Gergonne Line, the Gergonne Point, the Fletcher Point and the points of Eppstein, Griffith, Rigby and Nobbs as well. Amicable triangles and perfect circles have numerous and unusual interesting properties, and only a small part is presented in this article. Some of these results are still lacking of a rigorous mathematical proof; they only have been numerically confirmed. Key Words: triangle geometry, amicable triangles, perfect circles, Soddy Line, Gergonne Point, Gergonne Line, Nobbs Points MSC 2010: 51M04 1. Introduction This paper presents generalizations of several well known geometric objects, whose special cases are described in numerous sources. Despite of an intensive literature recherche, to the author's best knowledge these generalizations seem to be new. This is why references to this article contain only four positions, which I mainly used. All these considerations take their beginning from two triangles inscribed in any circle ac- companying to the reference triangle ABC. These triangles, which I called amicable triangles, allowed me to define some family of circles { perfect circles. On their base it was possible to build generalizations of geometrical objects like the Soddy and Gergonne Line, the Gergonne Point, the Fletcher Point and the points of Eppstein, Griffith, Rigby and Nobbs as as well as the pair of other points, which do not exist for the incircle. Amicable triangles and perfect circles have numerous and unusual interesting properties. I have been working on this theme for ten years and I put it together on over 200 pages of my elaborations. This article shows only a small part out of these properties. ISSN 1433-8157/$ 2.50 c 2013 Heldermann Verlag 54 M. Sejfried: Amicable Triangles and Perfect Circles 2. Amicable Triangles Given any reference triangle ABC. Let's join each of its vertices A, B and C with correspond- ing points S0, T0 and U0 lying on the opposite sides. So we obtain the three cevians AS0, BT0 and CU0. These cevians intersect at three points K0, L0 and M0 forming a triangle (cevianic triangle), which in special cases according to Ceva's Theorem degenerates to a single point. Let's now circumscribe the circle c(O0; r0) on the triangle K0L0M0 (Fig. 1). Thereafter, let's modify the points S0, T0 and U0 along the sides BC, AC and AB such that the circle c(O0; r0) becomes the incircle of the triangle ABC. There are exactly two such triples of points S, T and U as well as S1, T1 and U1 for which it is satisfied. B Uo Lo Oo So Mo l0 K A o To C Figure 1: The Golden Theorem When we set jBSj jCT j jAUj s = ; t = ; u = ; jSCj jTAj jUBj (1) jBS1j jCT1j jAU1j s1 = ; t1 = ; u1 = ; jS1Cj jT1Aj jU1Bj and ' for the golden mean we obtain a theorem which I called the Golden Theorem. Theorem 1. Golden Theorem: The triangles KLM and K1L1M1, which sides lie on six cevians passing pairwise through the vertices of the triangle ABC and meeting on the incircle of ABC, define ratios which satisfy p 6 1 ±1 + 5 stu = = = '±6: (2) s1t1u1 2 The same formula holds also for any ellipse inscribed in the triangle ABC. M. Sejfried: Amicable Triangles and Perfect Circles 55 B B e n i L y d d o S S1 U S1 L1 L U F k B2 L1 1 1 A1 A m L F C1 C1 A3 E G O C3 G E k Ox S 1 m U1 S Qx M K1 Q M H K1 U1 H A2 C2 D J D M1 J K l l1 K B3 A M1 A T B1 T T1 C B1 T1 C Figure 2: Amicable triangles in the incircle Figure 3: General amicable triangles The last remark follows by applying an affine transformation. The two triangles KLM and K1L1M1 are called amicable triangles. They mutually in- tersect on the cevians AA1, BB1 and CC1 called main cevians, which meet at the Gergonne Point Q (Fig. 2). There exist at least two proofs of the Golden Theorem (2). One originates from Vladimir Shelomovskii [2] and another from the young Chinese mathematician Yuming Li. The second proof | yet unpublished | is based on Ptolemy's Theorem and on harmonic quadri- laterals. Based on the Golden Theorem we get for s, t, u, s1, t1 and u1 the following equations: p p p p 3 + 5 a − b + c 3 + 5 rB 3 − 5 a − b + c 3 − 5 rB s = · = · ; s1 = · = · ; 2 a + b − c 2 rC 2 a + b − c 2 rC p p p p 3 + 5 b − c + a 3 + 5 rC 3 − 5 b − c + a 3 − 5 rC t = · = · ; t1 = · = · ; (3) 2 b + c − a 2 rA 2 b + c − a 2 rA p p p p 3 + 5 c − a + b 3 + 5 rA 3 − 5 c − a + b 3 − 5 rA u = · = · ; u1 = · = · ; 2 c + a − b 2 rB 2 c + a − b 2 rB where rA, rB and rc are the radii of three mutually tangent vertex circles of the triangle ABC. 56 M. Sejfried: Amicable Triangles and Perfect Circles We get then the coordinates of the points S, T , U, S1, T1, and U1: x 1 x p x S = p p 2(a + b − c) B + (3 + 5)(a − b + c) C yS (5 + 5)a − (1 + 5)(b − c) yB yC x 1 x p x T = p p 2(b + c − a) C + (3 + 5)(a + b − c) A (4) yT (5 + 5)b − (1 + 5)(c − a) yC yA x 1 x p x U = p p 2(a − b + c) A + (3 + 5)(b + c − a) B yU (5 + 5)c − (1 + 5)(a − b) yA yB x 1 x p x S1 = p p 2(a + b − c) B + (3 − 5)(a − b + c) C yS1 (5 − 5)a − (1 − 5)(b − c) yB yC x 1 x p x T1 = p p 2(b + c − a) C + (3 − 5)(a + b − c) A (5) yT1 (5 − 5)b − (1 − 5)(c − a) yC yA x 1 x p x U1 = p p 2(a − b + c) A + (3 − 5)(b + c − a) B yU1 (5 − 5)c − (1 − 5)(a − b) yA yB and resulting from above the coordinates of the points K, L, M, K1, L1, and M1: xK 1 = p p p p · yK 2 (7 + 3 5)bc + (3 + 5)(ab − c2) + (1 + 5)a2(4 + 2 5)b2 + 2ac p x x p x (7 + 3 5) a2 − (b−c)2 A + 2 b2 − (c−a)2 B + (3 + 5) c2 − (a−b)2 C yA yB yC xL 1 = p p p p · yL 2 (7 + 3 5)ac + (3 + 5)(bc − a2) + (1 + 5)b2(4 + 2 5)c2 + 2ab (6) p x p x x (7 + 3 5) b2 − (c−a)2 B + (3 + 5) a2 − (b−c)2 A + 2 c2 − (a−b)2 C yB yA yC xM 1 = p p p p · yM 2 (7 + 3 5)ab + (3 + 5)(ac − b2) + (1 + 5)c2(4 + 2 5)a2 + 2bc p x p x x (7 + 3 5) c2 − (a−b)2 C + (3 + 5) b2 − (c−a)2 B + 2 a2 − (b−c)2 A yC yB yA x 1 K1 = p p p p · 2 2 2 yK1 2 (7 − 3 5)ab + (3 − 5)(ac − b ) − ( 5 − 1)c (4 − 2 5)a + 2bc p x p x x (7 − 3 5) c2 − (a−b)2 C + (3 − 5) b2 − (c−a)2 B + 2 a2 − (b−c)2 A yC yB yA x 1 L1 = p p p p · 2 2 2 xL1 2 (7 − 3 5)bc + (3 − 5)(ab − c ) − ( 5 − 1)a (4 − 2 5)b + 2ac (7) p x p x x (7 − 3 5) a2 − (b−c)2 A + (3 − 5) c2 − (a−b)2 C + 2 b2 − (a−c)2 B yA yC yB x 1 M1 = p p p p · 2 2 2 xM1 2 (7 − 3 5)ac + (3 − 5)(bc − a )(4 − 2 5)c − ( 5 − 1)b + 2ab p x p x x (7 − 3 5) b2 − (a−c)2 B + (3 − 5) a2 − (b−c)2 A + 2 c2 − (a−b)2 C yB yA yC The amicable triangles based on the incircle have many interesting properties. Here are two of them: 2 2 2 2 2 2 k + l + m k1 + l1 + m1 1 1 1 1 1 1 = ; 2 + 2 + 2 = 2 + 2 + 2 (8) PKLM PK1L1M1 k l m k1 l1 m1 M. Sejfried: Amicable Triangles and Perfect Circles 57 where k, l, m, k1, l1, m1 are the respective lengths of the sides of the triangles KLM and K1L1M1 while PKLM and PK1L1M1 are the surface areas of these triangles. This result has been proved by V. Shelomovskii [2]. The above equations hold also for generalized amicable triangles. The segments A1S, A1S1, B1T , B1T1, C1U, and C1U1 have the following property | among numerous other properties: 1 1 1 1 1 1 + + = + + jA Sj jB T j jC Uj jA S j jB T j jC U j 1 1 1 1 1 1 1 1 1 (9) 1 1 1 1 1 1 2 + 2 + 2 = 2 + 2 + 2 jA1Sj jB1T j jC1Uj jA1S1j jB1T1j jC1U1j and resulting from these equations: jA Sj + jB T j + jC Uj jA Sj · jB T j · jC Uj 1 1 1 = 1 1 1 (10) jA1Sj + jB1T j + jC1Uj jA1Sj · jB1T j · jC1Uj I called the pairs of triples of numbers satisfying above and similar equations quadratic ter- tionals.
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