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10919Re: [EMHL] Feuerbach and pedal triangles

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  • Paul Yiu
    Dec 1, 2004
      Dear Eric,

      [ED]: I conjecture the following generalization:

      >Let O be the circumcenter and I the incenter
      >The perpendiculars from each excenter to the line joining the
      >Feuerbach point with the corresponding vertex of the pedal triangle
      >of any point on OI are concurrent.
      >
      >The following questions intrigue me:
      >
      >==> what is the locus of the perspectors ?

      Your conjecture is correct. The locus of the perspector is the Jerabek
      hyperbola of the excentral triangle.
      With reference to ABC, this has equation

      bc(b-c)x^2 + ca(c-a)y^2 + ab(a-b)z^2 = 0.

      Best regards
      Sincerely
      Paul



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