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Feuerbach and pedal triangles

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  • Eric Danneels
    Dear Hyacinthians, ... The perpendiculars from each excenter to the line joining the Feuerbach point with the corresponding vertex of the medial triangle are
    Message 1 of 3 , Dec 1, 2004
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      Dear Hyacinthians,

      in message 8533 I wrote:
      -----------------------------------------------------------------
      The perpendiculars from each excenter to the line joining the
      Feuerbach point with the corresponding vertex of the medial triangle
      are concurrent.
      -----------------------------------------------------------------
      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 ?
      ==> is OI the only set of points with this property?
      ==> if we replace the Feuerbach point by another point is there
      another line, or is this property unique for the Feuerbach point ?

      I suppose it all has something to do with orthopoles since the
      Feuerbach point is the orthopole of the OI-line, but I can't find
      out why

      Greetings from Bruges

      Eric
    • Paul Yiu
      Dear Eric, ... Your conjecture is correct. The locus of the perspector is the Jerabek hyperbola of the excentral triangle. With reference to ABC, this has
      Message 2 of 3 , Dec 1, 2004
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        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



        [Non-text portions of this message have been removed]
      • Paul Yiu
        Dear Eric, ... Apart from the OI line, there is also the line containing X(36) and its isogonal conjugate X(80). X(36) is the inversive image of I in the
        Message 3 of 3 , Dec 1, 2004
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          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 ?
          >==> is OI the only set of points with this property?

          Apart from the OI line, there is also the line containing X(36) and its
          isogonal conjugate X(80).
          X(36) is the inversive image of I in the circumcircle.

          Best regards
          Sincerely
          Paul



          [Non-text portions of this message have been removed]
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