Triangles Homothetic with Triangle ABC. Part 2

International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 c IJCDM Volume 2, 2017, pp.90-96 Received 1 May 2017. Published on-line 10 May 2017 web: http://www.journal-1.eu/ c The Author(s) This article is published with open access1. Triangles Homothetic with Triangle ABC. Part 2 Sava Grozdeva , Hiroshi Okumurab and Deko Dekovc 2 VUZF University of Finance, Business and Entrepreneurship, Gusla Street 1, 1618 Sofia, Bulgaria e-mail: [email protected] b Maebashi Gunma, 371-0123, Japan e-mail: [email protected] c Zahari Knjazheski 81, 6000 Stara Zagora, Bulgaria e-mail: [email protected] web: http://www.ddekov.eu/ a Abstract. By using the computer program ”Discoverer” we study triangles homothetic with the reference triangle ABC. Keywords. homothety, triangle geometry, remarkable point, computer discovered mathematics, Euclidean geometry, “Discoverer”. Mathematics Subject Classification (2010). 51-04, 68T01, 68T99. 1. Introduction We continue the investigation of triangles homothetic with the reference triangle ABC. For the first part of this papers see [7]. Theorems in this papers are discovered by the computer program ”Discoverer” created by the authors. We use barycentric coordinates. See [1]-[17]. The Kimberling points are denoted by X(n). We present a few problems related the topic. We encourage th students and researchers to solve them. 1This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. 2Corresponding author 90 Sava Grozdev, Hiroshi Okumura and Deko Dekov 91 2. Homothetic Triangles Theorem 1. Triangle ABC is homothetic with the Triangle T1 of Reflections of the Nine-Point Center in the Sidelines of the Medial triangle. The center of the homothety is the Circumcenter. The ratio of the homothety is 12 . Figure 1. Figure 1 illustrates Theorem 1. In figure 1, M aM bM c is the Medial triangle, N is the Nine-Point Center, Ra is the reflection of point N in the line M bM c, Rb is the reflection of point N in the line M cM a, Rc is the reflection of point N in the line M aM b, RaRbRc is the triangle of reflections of point N in the sidelines of triangle M aM bM c, • O is the Circumcenter. • • • • • • Triangles ABC and RaRbRc are homothetic and the center of the homothety is the Circumcenter. Proof. We leave to the reader the proof that triangles ABC and RaRbRc are homothetic with the Circumcenter as the center of the homothety. We will find the ratio of the homothety. We use barycentric coordinates. The Medial triangle is the cevian triangle of the Centroid. By using formula (3) in [5] we find the barycentric equation of the line M bM c as the line through points M b and M c as follows: −x + y + z = 0. By using formula (8) in [5] we find the equation of the line L through the Nine-point center N and perpendicular to line M bM c, as follows: L : (b2 − c2 )x + (c2 + 2a2 − b2 )y + (c2 − 2a2 − b2 )z = 0. By using formula (5) in [5] we find the intersection Q of the lines M bM c and L as follows: Q = (2, 1, 1). By using formula (15) in [5] we find the reflection Ra of point N in point Q, as follows: Ra = (b4 − 3b2 a2 − 2b2 c2 + c4 − 3c2 a2 + 2a4 , −b2 (−b2 + c2 + a2 ), −c2 (−c2 + a2 + b2 )). Triangles Homothetic with Triangle ABC. Part 2 92 By using the distance formula (9) in [5], we find the segments ORa and OA, and finally we obtain for the ratio: 1 ORa = . k= OA 2 This completes the proof.  We see that the triangle T1 in fact is the Euler triangle of the Circumcenter. Theorem 2. Triangle ABC is homothetic with the Triangle T2 of Reflections of the Orthocenter in the Sidelines of the Orthic triangle. The center of the homothety is the point X(24). The ratio of the homothety is (b2 + c2 − a2 )(c2 + a2 − b2 )(a2 + b2 − c2 ) 2a2 b2 c2 If triangle ABC is acute, then k > 0, if it is obtuse, then k < 0.  k= Figure 2. Figure 2 illustrates Theorem 2. In figure 2, • • • • • • H is the Orthocenter, HaHbHc is the Orthic triangle, Ra is the reflection of H in the line HbHc, Rb is the reflection of H in the line HcHa, Rc is the reflection of H in the line HaHb, RaRbRc is the Triangle of Reflections of point H in the side lines of triangle HaHbHc, O is the point X(24). Triangles ABC and RaRbRc are homothetic and the center of the homothety is the point X(24). Theorem 3. Triangle ABC is homothetic with the Triangle T3 of Reflections of the Circumcenter in the Sidelines of the Tangential triangle. The center of the homothety is the Circumcenter. The retio of the homothety is 2.  Figure 3 illustrates Theorem 3. In figure 3, • K is the Symmedian Point, • KaKbKv is the Tangential triangle, • Ra is the Reflection of the Circumcenter in the side line KbKc, Sava Grozdev, Hiroshi Okumura and Deko Dekov 93 Figure 3. Figure 4. • Rb is the Reflection of the Circumcenter in the side line KcKa, • Rc is the Reflection of the Circumcenter in the side line KaKb, • O is the Circumcenter. Triangles ABC and RaRbRc are homothetic and the center of the homothety is the Circumcenter. Theorem 4. Triangle ABC is homothetic with the Triangle T4 of Reflections of the Orthocenter in the Sidelines of the Antimedial Triangle. The center of the homothety is the Orthocenter. The ratio is 2. Figure 4 illustrates Theorem 4. In figure 4, • • • • • • H is the Orthocenter, G is the Centroid, P aP bP c is the Antimedial triangle, Ra is the Reflection of H in the side line P bP c, Rb is the Reflection of H in the side line P cP a, Rc is the Reflection of H in the side line P aP b, 94 Triangles Homothetic with Triangle ABC. Part 2 • RaRbRc is the Triangle of Reflections of H in the side lines of the Antimedial triangle. Triangles ABC and RaRbRc are homothetic and the center of the homothety is the Orthocenter. 3. Barycentric Coordinates via Homothety Now we are in position to find the barycentric coordinates of homothetic triangles. If triangles ABC and RaRbRc are homothetic under the homothety h(O, k) with center O and ratio k, then Ra = h(A), Rb = h(B) and Rc = h(C). We use the homothety formula (17) in [5]. Theorem 5. The barycentric coordinates of the Triangle T1 of the Reflections of the Nine-Point Center in the Sidelines of the Medial triangle are as follows: Ra = (3a2 b2 + 3a2 c2 − 2a4 + 2b2 c2 − b4 − c4 , b2 (c2 + a2 − b2 ), c2 (a2 + b2 − c2 )), Rb = (a2 , (b2 + c2 − a2 ), 3b2 c2 + 3a2 b2 − 2b4 + 2a2 c2 − a4 − c4 , c2 (a2 + b2 − c2 )), Rc = (a2 , (b2 + c2 − a2 ), b2 , (c2 + a2 − b2 ), 3a2 c2 + 3b2 c2 − 2c4 + 2a2 b2 − a4 − b4 ). Note that the same barycentric coordinates are given in [6]. Problem 3.1. Find the barycentric coordinates of triangles T2 to T4 in Theorems 2 to 4. Now we are also in position to find the barycentric coordinates of notable points of triangles homothetic with triangle ABC. We use the homothety formula (17) in [5]. Theorem 6. The barycentric coordinates of the Centroid GT of the Triangle T1 of the Reflections of the Nine-Point Center in the Sidelines of the Medial triangle are as follows: uGT 1 = 5a2 b2 + 5a2 c2 − 4a4 + 2b2 c2 − b4 − c4 vGT 1 = 5b2 c2 + 5a2 b2 − 4b4 + 2a2 c2 − a4 − c4 wGT 1 = 5a2 c2 + 5b2 c2 − 4c4 + 2a2 b2 − a4 − b4 Problem 3.2. Find the barycentric coordinates of the following notable points of triangle T1 in Theorems 1: Centroid, Incenter, Circumcenter, Orthocenter. Problem 3.3. Find the barycentric coordinates of the following notable points of triangles T2 −T4 in Theorems 2-4: Centroid, Incenter, Circumcenter, Orthocenter. 4. Kimberling Points of Triangle T1 We have investigated 195 notable points of triangle T1 . Of these 42 are Kimberling points and the rest of 153 points ane new points, not available in Kimberling [10]. Below is a part of the Kimberlin points. See also the Supplementary material. Table 1 gives notable points of Triangle T1 in terms of the notable points of the Reference Triangle ABC that are Kimberling points X(n). The ”Disciverer” gives us the opportunity to add a number of new properties to the properties available in [10]. For example: Sava Grozdev, Hiroshi Okumura and Deko Dekov Notable Points of triangle T1 1 2 3 4 5 6 Incenter Centroid Circumcenter Orthocenter Nine-Point Center Symmedian Point 95 Notable Points of Triangle ABC X(1385) X(549) X(3) X(5) X(140) X(182) Table 1. Figure 5. Theorem 7. The Centroid of the Triangle of Reflections of the Nine-Point Center in the Sidelines of the Medial Triangle (Point X(549 in [10])) is the Center of the Orthocentroidal Circle of the Medial Triangle. Figure 5 illustrates Theorem 7. In figure 5 • • • • • M aM bM c is the Medial triangle, G is the Centroid of the Medial triangle, H is the Orthocenter of the Medial triangle, c is the Orthocentroidal circle of the Medial triangle, O is the center of circle c, that is, the point X(549) = Centroid of triangle T1 . 5. New Points of Triangle T1 We have found 153 new notable points of Triangle T1 . By using the homothety h1 we can fing the barycentric coordinates of these points and by using the ”Discoverer” we can find a number of properties of these points. For example: Theorem 8. The Gergonne Point of the Triangle T1 is the Midpoint of the Circumcenter and the Gergonne Point. Problem 5.1. Find the barycentric coordinates of the Gergonne Point of Triangle T1 . 96 Triangles Homothetic with Triangle ABC. Part 2 Supplementary Material The enclosed supplementary material contains theorems related to the topic. Acknowledgement The authors are grateful to Professor René Grothmann for his wonderful computer program C.a.R. http://car.rene-grothmann.de/doc_en/index.html. See also http://www.journal-1.eu/2016-1/Grothmann-CaR-pp.45-61.pdf. References [1] César Lozada, Index of triangles referenced in ETC. http://faculty.evansville.edu/ ck6/encyclopedia/IndexOfTrianglesReferencedInETC.html. [2] Francisco Javier Garcı́a Capitán, Barycentric Coordinates, International Journal of Computer Discovered Mathematics, 2015, vol.0, no.0, 32-48. http://www.journal-1.eu/2015/ 01/Francisco-Javier-Barycentric-Coordinates-pp.32-48.pdf. [3] Pierre Douillet, Translation of the Kimberling’s Glossary into barycentrics, 2012, v48, http: //www.douillet.info/~douillet/triangle/Glossary.pdf. [4] S. Grozdev and D. Dekov, A Survey of Mathematics Discovered by Computers, International Journal of Computer Discovered Mathematics, 2015, vol.0, no.0, 3-20. http: //www.journal-1.eu/2015/01/Grozdev-Dekov-A-Survey-pp.3-20.pdf. 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