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Re: A dodecagon

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  • rhutson2
    Dear Francisco and Antreas, This perspector S is also the crosssum of X(6) and X(1350), the crosspoint of X(2) and X(3424), and also lies on line X(4)X(32).
    Message 1 of 3 , Jan 31, 2013
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      Dear Francisco and Antreas,

      This perspector S is also the crosssum of X(6) and X(1350), the crosspoint of X(2) and X(3424), and also lies on line X(4)X(32).
      The isotomic conjugate of this circumconic is a line that meets the line at infinity at the isogonal conjugate of Psi(X(76), X(20)), which is also the circumcircle intercept of line X(6)X(1297) (besides X(1297)).

      The center of this circumconic is the X(2)-Ceva conjugate of S, and so has barycentric coordinates:

      f(a,b,c)( -f(a,b,c) + f(b,c,a) + f(c,a,b)) : :,

      where f(a,b,c) = 3a^4 + (b^2 - c^2)^2

      Best regards,
      Randy

      --- In Hyacinthos@yahoogroups.com, "Francisco Javier" wrote:
      >
      > Dear Antreas,
      >
      > your conic is always homothetic to the circumconic with perspector
      >
      > S={3 a^4 + b^4 - 2 b^2 c^2 + c^4, a^4 + 3 b^4 - 2 a^2 c^2 + c^4,
      > a^4 - 2 a^2 b^2 + b^4 + 3 c^4}.
      >
      > This point S divides de segment GK (G=centroid, K=symmedian point) in the ratio (-2/3)(cotw^2), where w is the Brocard angle of the triangle.
      >
      > Best regards,
      >
      > Francisco Javier.
      >
      > --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
      > >
      > >
      > > http://anthrakitis.blogspot.gr/2013/01/conics-centered-at-o.html
      > >
      > > aph
      > >
      >
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