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7221Variation on the Van Lamoen-Grinberg-Wolk-transform

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  • efn4900
    Jun 1, 2003
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      Dear Hyacynthians,

      looking for variations of the Van Lamoen-Grinberg-Wolk-transform I
      discovered the following properties:

      Consider a point P and a triangle ABC.
      Let A'B'C' be the circumcevian triangle of P with respect to ABC.
      Let A°, B° and C° be the circumcenters of the triangles PB'C', PC'A'
      and PA'B'.

      ==> The lines AA°, BB° and CC° pass through a point S on the
      circumcircle of ABC.


      I found the following relationships using the ETC

      P = Incenter X(1) ==> S = X(104)
      P = Centroid X(2) ==> S = X(98)
      P = Circumcenter X(3) ==> S = X(74)
      P = Orthocenter X(4) ==> S = undetermined
      P = Euler-center X(5) ==> S = X(1141)
      P = Lemoine point X(6) ==> S = X(74)
      P = Isogonal conjugate of Euler-center X(54) ==> S = X(74)


      If we consider ABC as the circumcevian triangle of P with respect to
      A'B'C' we have

      Let A*, B* and C* be the circumcenters of the triangles PBC, PCA and
      PAB.

      ==> The lines A'A*, B'B* and C'C* pass through a point T on the
      circumcircle of ABC,

      I found the following using the ETC

      P = Incenter X(1) ==> T = undetermined
      P = Centroid X(2) ==> T = X(1296)
      P = Circumcenter X(3) ==> T = X(110)
      P = Orthocenter X(4) ==> T = X(110)
      P = Euler-center X(5) ==> T = X(1291)
      P = Lemoine point X(6) ==> T = X(1296)
      P = Isogonal conjugate of Euler-center X(54) ==> T = X(1291)

      It seems that isogonal conjugates map to the same point.

      Are these properties known ?

      Greetings from Belgium

      Eric Danneels
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