DOI QR코드

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Isotomic and Isogonal Conjugates Tangent Lines of Lines at Vertices of Triangle

  • 투고 : 2017.02.08
  • 심사 : 2017.03.25
  • 발행 : 2017.03.30

초록

In this paper we consider the two tangent lines of isogonal and isotomic conjugates of the line at both vertices of a given triangle. We find the necessary and sufficient condition for the two tangent lines of isogonal or isotomic conjugates of the line at both vertices and the median line to be concurrent. We also prove that every line whose isogonal conjugate tangent lines at both vertices are concurrent with the median line intersects at a unique point. Moreover, we show that the three intersection points correspond to the vertices of triangle are collinear.

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참고문헌

  1. A. Canton, L. Fernandez-Jambrina, and E. R. Maria. "eometric characteristics of conics in Bezier form", Comput. Aided Design, Vol. 43, pp. 1413-1421, 2011. https://doi.org/10.1016/j.cad.2011.08.025
  2. L. Fang, "3 approximation of conic sections by quintic polynomial curves", Comput. Aided Geom. D., Vol. 16, pp. 755-766, 1999. https://doi.org/10.1016/S0167-8396(99)00017-5
  3. R. T. Farouki, "onic approximation of conic offsets", Lancet Infect. Dis., Vol. 23, pp. 301-313, 1997.
  4. M. S. Floater, "An O($h^{2n}$) Hermite approximation for conic sections", Comput. Aided Geom. D., Vol. 14, pp. 135-151, 1997. https://doi.org/10.1016/S0167-8396(96)00025-8
  5. H. Y. Park and Y. J. Ahn, "Isoparametric curve of quadratic F-Bezier curve", J. Chosun Natural Sci., Vol. 6, pp. 46-52, 2013. https://doi.org/10.13160/ricns.2013.6.1.046
  6. G. J. Wang and G. Z. Wang, "The rational cubic Bezier representation of conics", Comput. Aided Geom. D., Vol. 9, pp. 447-455, 1992. https://doi.org/10.1016/0167-8396(92)90043-O
  7. M. Dalcin, "Isotomic inscribed triangles and their residuals", Forum Geometricorum, Vol. 3, pp. 125-134, 2003.
  8. A. P. Hatzipolakis, F. van Lamoen, B. Wolk, and P. Yiu, "Concurrency of four Euler lines", Forum Geometricorum, Vol. 1, pp. 59-68, 2001.
  9. K. R. S. Sastry, "Triangles with special isotomic conjugate pairs", Forum Geometricorum, Vol. 4, pp. 73-80, 2004.
  10. P. Yiu, "The uses of homogeneous barycentric coordinates in plane Euclidean geometry", International Journal of Mathematical Education in Science and Technology, Vol. 31, pp. 569-578, 2000. https://doi.org/10.1080/002073900412679
  11. K. Dean and F. van Lamoen, "Geometric construction of reciprocal conjugation", Forum Geometricorum, Vol. 1, pp. 115-120, 2001.
  12. A. V. Akopyan, "Conjugation of lines with respect to a triangle", Journal of Classical Geometry, Vol. 1, pp. 23-31, 2012.
  13. P. Yiu, "Conic construction of a triangle from the feet of its angle bisectors", Journal for Geometry and Graphics, Vol. 12, pp. 171-182, 2008.
  14. M. Y. Seo, "Tangent line of isotomic conjugate", MS diss., Chosun University, 2016.
  15. P. Yiu, "Introduction to the geometry of the triangle", 2001.
  16. C. R. Yun and Y. J. Ahn, "Isogonal and isotomic conjugates of quadratic rational Bezier curves", Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, Vol. 22, pp. 25-34, 2015.
  17. R. Lee and Y. J. Ahn, "Geometric shape analysis for convolution curve of two compatible quadratic Bezier curves", J. Comput. Appl. Math., Vol. 288, pp. 141-150, 2015. https://doi.org/10.1016/j.cam.2015.04.012