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Re: [EMHL] Parallel tripolars

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  • Bernard Gibert
    Dear Antreas and Steve, ... This is only a special case of the more general theorem : If W (p,q,r) is the pole of the isoconjugation P(x,y,z) ----
    Message 1 of 3 , Apr 17, 2001
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      Dear Antreas and Steve,

      >> Steve wrote:
      >>
      >>> Mineur ---
      >>>
      >>> The 12 point cubic (isogonic with pivot K) is the locus of inverse points
      >>> whose axes of perspective are parallel.
      >>
      This is only a special case of the more general theorem :

      If W (p,q,r) is the pole of the isoconjugation
      P(x,y,z) ----> P'(p/x,q/y,r/z),

      then the locus of P such that the tripolars of P and P' are parallel is the
      pivotal cubic with pivot W i.e. pole = pivot !

      Equation (barys) : px(ry^2 - qz^2) + cyclic = 0.

      When W = K, you have the isogonality with the 12 point cubic.
      When W = G, you have the isotomy with the union of the 3 medians.

      Remark : there is only one circular such cubic for W = X67 (isotomic of the
      Droussent pivot X316).

      Best regards

      Bernard
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