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19950Forum Geometricorum

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  • Forum Geometricorum
    Apr 1, 2011
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      The following paper has been published in Forum Geometricorum. It can be viewed at


      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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