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Forum Geometricorum

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  • Forum Geometricorum
    The following paper has been published in Forum Geometricorum. It can be viewed at http://forumgeom.fau.edu/FG2011volume11/FG201109index.html The editors Forum
    Message 1 of 476 , Apr 1, 2011
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      The following paper has been published in Forum Geometricorum. It can be viewed at

      http://forumgeom.fau.edu/FG2011volume11/FG201109index.html

      The editors
      Forum Geometricorum

      Peter J. C. Moses and Clark Kimberling, Perspective isoconjugate triangle pairs, Hofstadter pairs, and crosssums on the nine-point circle,
      Forum Geometricorum, 11 (2011) 83--93.

      Abstract. The r-Hofstadter triangle and the (1-r)-Hofstadter triangle are proved perspective, and homogeneous trilinear coordinates are found for the perspector. More generally, given a triangle DEF inscribed in a reference triangle ABC, triangles A'B'C' and A''B''C'' derived in a certain manner from DEF are perspective to each other and to ABC. Trilinears for the three perspectors, denoted by P*, P_1, P_2 are found (Theorem 1) and used to prove that these three points are collinear. Special cases include (Theorems 4 and 5) this: if X and X' are an antipodal pair on the circumcircle, then the perspector P* = X \oplus X', where \oplus denotes crosssum, is on the nine-point circle. Taking X to be successively the vertices of a triangle DEF inscribed in the circumcircle thus yields a triangle D'E'F' inscribed in the nine-point circle. For example, if DEF is the circumtangential triangle, then D'E'F' is an equilateral triangle.


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    • 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
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        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.


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