Loading ...
Sorry, an error occurred while loading the content.

Forum Geometricorum

Expand Messages
  • ForumGeom
    The following paper has been published in Forum Geometricorum. It can be viewed at http://forumgeom.fau.edu/FG2009volume9/FG200931index.html The editors Forum
    Message 1 of 476 , Dec 22, 2009
    • 0 Attachment
      The following paper has been published in Forum Geometricorum. It can be
      viewed at

      http://forumgeom.fau.edu/FG2009volume9/FG200931index.html

      The editors
      Forum Geometricorum
      ==============================================
      Antreas P Hatzipolakis and Paul Yiu, Reflections in triangle geometry,
      Forum Geometricorum, 9 (2009) 301--348.

      On the 10th Anniversary of Hyacinthos

      Abstract. This paper is a survey of results on reflections in triangle
      geometry. We work with homogeneous barycentric coordinates with reference
      to a given triangle ABC and establish various concurrency and perspectivity
      results related to triangles formed by reflections, in particular the
      reflection triangle P^(a)P^(b)P^(c) of a point $P$ in the sidelines of ABC,
      and the triangle of reflections A^(a)B^(b)C^(c) of the vertices of ABC in
      their respective opposite sides. We also consider triads of concurrent
      circles related to these reflections. In this process, we obtain a number
      of interesting triangle centers with relatively simple coordinates. While
      most of these triangle centers have been catalogued in Kimberling's
      Encyclopedia of Triangle Centers (ETC), there are a few interesting new
      ones. We give additional properties of known triangle centers related to
      reflections, and in a few cases, exhibit interesting correspondences of
      cubic curves catalogued in Gibert's Catalogue of Triangle Cubics (CTC).


      [Non-text portions of this message have been removed]
    • forumgeom forumgeom
      The following paper has been published in Forum Geometricorum. It can be viewed at http://forumgeom.fau.edu/FG2013volume13/FG201309ndex.html The editors Forum
      Message 476 of 476 , Apr 16, 2013
      • 0 Attachment
        The following paper has been published in Forum Geometricorum. It can be viewed at

        http://forumgeom.fau.edu/FG2013volume13/FG201309ndex.html

        The editors
        Forum Geometricorum

        Paul Yiu, On the conic through the intercepts of the three lines through the centroid and the intercepts of a given line,
        Forum Geometricorum, 13 (2013) 87--102.

        Abstract. Let L be a line intersecting the sidelines of triangle ABC at X, Y, Z respectively. The lines joining these intercepts to the centroid give rise to six more intercepts on the sidelines which lie on a conic Q(L,G). We show that this conic (i) degenerates in a pair of lines if L is tangent to the Steiner inellipse, (ii) is a parabola if L is tangent to the ellipse containing the trisection points of the sides, (iii) is a rectangular hyperbola if L is tangent to a circle C_G with center G. We give a ruler and compass construction of the circle C_G. Finally, we also construct the two lines each with the property that the conic Q(L,G) is a circle.


        [Non-text portions of this message have been removed]
      Your message has been successfully submitted and would be delivered to recipients shortly.