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Parallelogic triangles [corrected]

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  • Antreas
    Let T = ABC be a triangle and L a line intersecting the sidelines BC,CA,AB of ABC at A , B , C , resp. Description of a transformation f of T in f(T): Let f(a)
    Message 1 of 4 , Jan 25, 2013
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      Let T = ABC be a triangle and L a line intersecting
      the sidelines BC,CA,AB of ABC at A', B', C', resp.

      Description of a transformation f of T in f(T):

      Let f(a) be the reflection of the perpendicular to BC at A'
      in the line L, and similarly f(b), f(c).

      Denote: f(T) = the triangle bounded by the lines f(a), f(b), f(c).

      The triangles T, f(T) are parallelogic and let x,y be the parallelogic centers.

      We have ff(T) = T and f(x) = y and f(y) = x
      [==> ff(x) = f(y) = x and ff(y) = f(x) = y]

      Which are the coordinates of x,y ?

      APH
    • Francisco Javier
      If L is the trilinear polar of P=(x:y:z) and X=parallelogic center center of T with respect f(T) Y=parallelogic center center of T with respect f(T) we have
      Message 2 of 4 , Jan 25, 2013
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        If L is the trilinear polar of P=(x:y:z) and
        X=parallelogic center center of T with respect f(T)
        Y=parallelogic center center of T with respect f(T)

        we have known points for X:

        for P=X1, X=X953,
        for P=X4, X=X477
        for P=X6, X=X2698
        for P=X7, X=X2724
        for P=X8, X=X2734

        Coordinates of isotomic conjugate of X and coordinates of Y are below.

        Best regards,

        Francisco Javier.

        ----------------

        Coordinates of isotomic conjugate of X are

        {a^2 c^2 x^2 y^2 - b^2 c^2 x^2 y^2 - c^4 x^2 y^2 +
        4 b^2 c^2 x^2 y z - 2 a^2 c^2 x y^2 z - 2 b^2 c^2 x y^2 z +
        2 c^4 x y^2 z + a^2 b^2 x^2 z^2 - b^4 x^2 z^2 - b^2 c^2 x^2 z^2 -
        2 a^2 b^2 x y z^2 + 2 b^4 x y z^2 - 2 b^2 c^2 x y z^2 +
        a^2 b^2 y^2 z^2 - b^4 y^2 z^2 + a^2 c^2 y^2 z^2 +
        2 b^2 c^2 y^2 z^2 - c^4 y^2 z^2, -a^2 c^2 x^2 y^2 +
        b^2 c^2 x^2 y^2 - c^4 x^2 y^2 - 2 a^2 c^2 x^2 y z -
        2 b^2 c^2 x^2 y z + 2 c^4 x^2 y z + 4 a^2 c^2 x y^2 z -
        a^4 x^2 z^2 + a^2 b^2 x^2 z^2 + 2 a^2 c^2 x^2 z^2 +
        b^2 c^2 x^2 z^2 - c^4 x^2 z^2 + 2 a^4 x y z^2 - 2 a^2 b^2 x y z^2 -
        2 a^2 c^2 x y z^2 - a^4 y^2 z^2 + a^2 b^2 y^2 z^2 -
        a^2 c^2 y^2 z^2, -a^4 x^2 y^2 + 2 a^2 b^2 x^2 y^2 - b^4 x^2 y^2 +
        a^2 c^2 x^2 y^2 + b^2 c^2 x^2 y^2 - 2 a^2 b^2 x^2 y z +
        2 b^4 x^2 y z - 2 b^2 c^2 x^2 y z + 2 a^4 x y^2 z -
        2 a^2 b^2 x y^2 z - 2 a^2 c^2 x y^2 z - a^2 b^2 x^2 z^2 -
        b^4 x^2 z^2 + b^2 c^2 x^2 z^2 + 4 a^2 b^2 x y z^2 - a^4 y^2 z^2 -
        a^2 b^2 y^2 z^2 + a^2 c^2 y^2 z^2}

        Coordinates of Y are

        {(y - z) (-a^4 c^2 x^4 y^2 + 2 a^2 b^2 c^2 x^4 y^2 -
        b^4 c^2 x^4 y^2 + 2 a^2 c^4 x^4 y^2 + 2 b^2 c^4 x^4 y^2 -
        c^6 x^4 y^2 - a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 -
        b^4 c^2 x^3 y^3 + c^6 x^3 y^3 - a^6 x^4 y z + 3 a^4 b^2 x^4 y z -
        3 a^2 b^4 x^4 y z + b^6 x^4 y z + 3 a^4 c^2 x^4 y z -
        2 a^2 b^2 c^2 x^4 y z - b^4 c^2 x^4 y z - 3 a^2 c^4 x^4 y z -
        b^2 c^4 x^4 y z + c^6 x^4 y z + a^6 x^3 y^2 z -
        3 a^4 b^2 x^3 y^2 z + 3 a^2 b^4 x^3 y^2 z - b^6 x^3 y^2 z -
        2 a^4 c^2 x^3 y^2 z - 6 a^2 b^2 c^2 x^3 y^2 z +
        8 b^4 c^2 x^3 y^2 z + a^2 c^4 x^3 y^2 z - 7 b^2 c^4 x^3 y^2 z +
        5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
        b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
        c^6 x^2 y^3 z - a^4 b^2 x^4 z^2 + 2 a^2 b^4 x^4 z^2 -
        b^6 x^4 z^2 + 2 a^2 b^2 c^2 x^4 z^2 + 2 b^4 c^2 x^4 z^2 -
        b^2 c^4 x^4 z^2 + a^6 x^3 y z^2 - 2 a^4 b^2 x^3 y z^2 +
        a^2 b^4 x^3 y z^2 - 3 a^4 c^2 x^3 y z^2 -
        6 a^2 b^2 c^2 x^3 y z^2 - 7 b^4 c^2 x^3 y z^2 +
        3 a^2 c^4 x^3 y z^2 + 8 b^2 c^4 x^3 y z^2 - c^6 x^3 y z^2 +
        a^4 b^2 x^2 y^2 z^2 - 2 a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 +
        a^4 c^2 x^2 y^2 z^2 + 20 a^2 b^2 c^2 x^2 y^2 z^2 -
        b^4 c^2 x^2 y^2 z^2 - 2 a^2 c^4 x^2 y^2 z^2 -
        b^2 c^4 x^2 y^2 z^2 + c^6 x^2 y^2 z^2 - a^6 x y^3 z^2 +
        2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
        4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
        b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 +
        5 a^4 b^2 x^2 y z^3 - 4 a^2 b^4 x^2 y z^3 - b^6 x^2 y z^3 -
        4 a^2 b^2 c^2 x^2 y z^3 + 2 b^4 c^2 x^2 y z^3 -
        b^2 c^4 x^2 y z^3 - a^6 x y^2 z^3 - 4 a^4 b^2 x y^2 z^3 +
        5 a^2 b^4 x y^2 z^3 + 2 a^4 c^2 x y^2 z^3 -
        4 a^2 b^2 c^2 x y^2 z^3 - a^2 c^4 x y^2 z^3 + a^6 y^3 z^3 -
        a^2 b^4 y^3 z^3 + 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3), (x -
        z) (a^4 c^2 x^3 y^3 - 2 a^2 b^2 c^2 x^3 y^3 + b^4 c^2 x^3 y^3 -
        c^6 x^3 y^3 + a^4 c^2 x^2 y^4 - 2 a^2 b^2 c^2 x^2 y^4 +
        b^4 c^2 x^2 y^4 - 2 a^2 c^4 x^2 y^4 - 2 b^2 c^4 x^2 y^4 +
        c^6 x^2 y^4 + a^4 c^2 x^3 y^2 z + 4 a^2 b^2 c^2 x^3 y^2 z -
        5 b^4 c^2 x^3 y^2 z - 2 a^2 c^4 x^3 y^2 z + 4 b^2 c^4 x^3 y^2 z +
        c^6 x^3 y^2 z + a^6 x^2 y^3 z - 3 a^4 b^2 x^2 y^3 z +
        3 a^2 b^4 x^2 y^3 z - b^6 x^2 y^3 z - 8 a^4 c^2 x^2 y^3 z +
        6 a^2 b^2 c^2 x^2 y^3 z + 2 b^4 c^2 x^2 y^3 z +
        7 a^2 c^4 x^2 y^3 z - b^2 c^4 x^2 y^3 z - a^6 x y^4 z +
        3 a^4 b^2 x y^4 z - 3 a^2 b^4 x y^4 z + b^6 x y^4 z +
        a^4 c^2 x y^4 z + 2 a^2 b^2 c^2 x y^4 z - 3 b^4 c^2 x y^4 z +
        a^2 c^4 x y^4 z + 3 b^2 c^4 x y^4 z - c^6 x y^4 z +
        a^4 b^2 x^3 y z^2 - 2 a^2 b^4 x^3 y z^2 + b^6 x^3 y z^2 +
        4 a^2 b^2 c^2 x^3 y z^2 + 4 b^4 c^2 x^3 y z^2 -
        5 b^2 c^4 x^3 y z^2 - a^6 x^2 y^2 z^2 + 2 a^4 b^2 x^2 y^2 z^2 -
        a^2 b^4 x^2 y^2 z^2 + a^4 c^2 x^2 y^2 z^2 -
        20 a^2 b^2 c^2 x^2 y^2 z^2 - b^4 c^2 x^2 y^2 z^2 +
        a^2 c^4 x^2 y^2 z^2 + 2 b^2 c^4 x^2 y^2 z^2 - c^6 x^2 y^2 z^2 -
        a^4 b^2 x y^3 z^2 + 2 a^2 b^4 x y^3 z^2 - b^6 x y^3 z^2 +
        7 a^4 c^2 x y^3 z^2 + 6 a^2 b^2 c^2 x y^3 z^2 +
        3 b^4 c^2 x y^3 z^2 - 8 a^2 c^4 x y^3 z^2 - 3 b^2 c^4 x y^3 z^2 +
        c^6 x y^3 z^2 + a^6 y^4 z^2 - 2 a^4 b^2 y^4 z^2 +
        a^2 b^4 y^4 z^2 - 2 a^4 c^2 y^4 z^2 - 2 a^2 b^2 c^2 y^4 z^2 +
        a^2 c^4 y^4 z^2 + a^4 b^2 x^3 z^3 - b^6 x^3 z^3 -
        2 a^2 b^2 c^2 x^3 z^3 + b^2 c^4 x^3 z^3 - 5 a^4 b^2 x^2 y z^3 +
        4 a^2 b^4 x^2 y z^3 + b^6 x^2 y z^3 + 4 a^2 b^2 c^2 x^2 y z^3 -
        2 b^4 c^2 x^2 y z^3 + b^2 c^4 x^2 y z^3 + a^6 x y^2 z^3 +
        4 a^4 b^2 x y^2 z^3 - 5 a^2 b^4 x y^2 z^3 - 2 a^4 c^2 x y^2 z^3 +
        4 a^2 b^2 c^2 x y^2 z^3 + a^2 c^4 x y^2 z^3 - a^6 y^3 z^3 +
        a^2 b^4 y^3 z^3 - 2 a^2 b^2 c^2 y^3 z^3 + a^2 c^4 y^3 z^3), (x -
        y) (-a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 - b^4 c^2 x^3 y^3 +
        c^6 x^3 y^3 - a^4 c^2 x^3 y^2 z - 4 a^2 b^2 c^2 x^3 y^2 z +
        5 b^4 c^2 x^3 y^2 z + 2 a^2 c^4 x^3 y^2 z - 4 b^2 c^4 x^3 y^2 z -
        c^6 x^3 y^2 z + 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
        b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
        c^6 x^2 y^3 z - a^4 b^2 x^3 y z^2 + 2 a^2 b^4 x^3 y z^2 -
        b^6 x^3 y z^2 - 4 a^2 b^2 c^2 x^3 y z^2 - 4 b^4 c^2 x^3 y z^2 +
        5 b^2 c^4 x^3 y z^2 + a^6 x^2 y^2 z^2 - a^4 b^2 x^2 y^2 z^2 -
        a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 - 2 a^4 c^2 x^2 y^2 z^2 +
        20 a^2 b^2 c^2 x^2 y^2 z^2 - 2 b^4 c^2 x^2 y^2 z^2 +
        a^2 c^4 x^2 y^2 z^2 + b^2 c^4 x^2 y^2 z^2 - a^6 x y^3 z^2 +
        2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
        4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
        b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 -
        a^6 x^2 y z^3 + 8 a^4 b^2 x^2 y z^3 - 7 a^2 b^4 x^2 y z^3 +
        3 a^4 c^2 x^2 y z^3 - 6 a^2 b^2 c^2 x^2 y z^3 +
        b^4 c^2 x^2 y z^3 - 3 a^2 c^4 x^2 y z^3 - 2 b^2 c^4 x^2 y z^3 +
        c^6 x^2 y z^3 - 7 a^4 b^2 x y^2 z^3 + 8 a^2 b^4 x y^2 z^3 -
        b^6 x y^2 z^3 + a^4 c^2 x y^2 z^3 - 6 a^2 b^2 c^2 x y^2 z^3 +
        3 b^4 c^2 x y^2 z^3 - 2 a^2 c^4 x y^2 z^3 - 3 b^2 c^4 x y^2 z^3 +
        c^6 x y^2 z^3 + a^6 y^3 z^3 - a^2 b^4 y^3 z^3 +
        2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3 - a^4 b^2 x^2 z^4 +
        2 a^2 b^4 x^2 z^4 - b^6 x^2 z^4 + 2 a^2 b^2 c^2 x^2 z^4 +
        2 b^4 c^2 x^2 z^4 - b^2 c^4 x^2 z^4 + a^6 x y z^4 -
        a^4 b^2 x y z^4 - a^2 b^4 x y z^4 + b^6 x y z^4 -
        3 a^4 c^2 x y z^4 - 2 a^2 b^2 c^2 x y z^4 - 3 b^4 c^2 x y z^4 +
        3 a^2 c^4 x y z^4 + 3 b^2 c^4 x y z^4 - c^6 x y z^4 -
        a^6 y^2 z^4 + 2 a^4 b^2 y^2 z^4 - a^2 b^4 y^2 z^4 +
        2 a^4 c^2 y^2 z^4 + 2 a^2 b^2 c^2 y^2 z^4 - a^2 c^4 y^2 z^4)}

        --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
        >
        > Let T = ABC be a triangle and L a line intersecting
        > the sidelines BC,CA,AB of ABC at A', B', C', resp.
        >
        > Description of a transformation f of T in f(T):
        >
        > Let f(a) be the reflection of the perpendicular to BC at A'
        > in the line L, and similarly f(b), f(c).
        >
        > Denote: f(T) = the triangle bounded by the lines f(a), f(b), f(c).
        >
        > The triangles T, f(T) are parallelogic and let x,y be the parallelogic centers.
        >
        > We have ff(T) = T and f(x) = y and f(y) = x
        > [==> ff(x) = f(y) = x and ff(y) = f(x) = y]
        >
        > Which are the coordinates of x,y ?
        >
        > APH
        >
      • Francisco Javier
        Sorry, I meant Y = parallelogic center center of f(T) with respect T
        Message 3 of 4 , Jan 25, 2013
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          Sorry, I meant

          Y = parallelogic center center of f(T) with respect T

          --- In Hyacinthos@yahoogroups.com, "Francisco Javier" wrote:
          >
          > If L is the trilinear polar of P=(x:y:z) and
          > X=parallelogic center center of T with respect f(T)
          > Y=parallelogic center center of T with respect f(T)
          >
          > we have known points for X:
          >
          > for P=X1, X=X953,
          > for P=X4, X=X477
          > for P=X6, X=X2698
          > for P=X7, X=X2724
          > for P=X8, X=X2734
          >
          > Coordinates of isotomic conjugate of X and coordinates of Y are below.
          >
          > Best regards,
          >
          > Francisco Javier.
          >
          > ----------------
          >
          > Coordinates of isotomic conjugate of X are
          >
          > {a^2 c^2 x^2 y^2 - b^2 c^2 x^2 y^2 - c^4 x^2 y^2 +
          > 4 b^2 c^2 x^2 y z - 2 a^2 c^2 x y^2 z - 2 b^2 c^2 x y^2 z +
          > 2 c^4 x y^2 z + a^2 b^2 x^2 z^2 - b^4 x^2 z^2 - b^2 c^2 x^2 z^2 -
          > 2 a^2 b^2 x y z^2 + 2 b^4 x y z^2 - 2 b^2 c^2 x y z^2 +
          > a^2 b^2 y^2 z^2 - b^4 y^2 z^2 + a^2 c^2 y^2 z^2 +
          > 2 b^2 c^2 y^2 z^2 - c^4 y^2 z^2, -a^2 c^2 x^2 y^2 +
          > b^2 c^2 x^2 y^2 - c^4 x^2 y^2 - 2 a^2 c^2 x^2 y z -
          > 2 b^2 c^2 x^2 y z + 2 c^4 x^2 y z + 4 a^2 c^2 x y^2 z -
          > a^4 x^2 z^2 + a^2 b^2 x^2 z^2 + 2 a^2 c^2 x^2 z^2 +
          > b^2 c^2 x^2 z^2 - c^4 x^2 z^2 + 2 a^4 x y z^2 - 2 a^2 b^2 x y z^2 -
          > 2 a^2 c^2 x y z^2 - a^4 y^2 z^2 + a^2 b^2 y^2 z^2 -
          > a^2 c^2 y^2 z^2, -a^4 x^2 y^2 + 2 a^2 b^2 x^2 y^2 - b^4 x^2 y^2 +
          > a^2 c^2 x^2 y^2 + b^2 c^2 x^2 y^2 - 2 a^2 b^2 x^2 y z +
          > 2 b^4 x^2 y z - 2 b^2 c^2 x^2 y z + 2 a^4 x y^2 z -
          > 2 a^2 b^2 x y^2 z - 2 a^2 c^2 x y^2 z - a^2 b^2 x^2 z^2 -
          > b^4 x^2 z^2 + b^2 c^2 x^2 z^2 + 4 a^2 b^2 x y z^2 - a^4 y^2 z^2 -
          > a^2 b^2 y^2 z^2 + a^2 c^2 y^2 z^2}
          >
          > Coordinates of Y are
          >
          > {(y - z) (-a^4 c^2 x^4 y^2 + 2 a^2 b^2 c^2 x^4 y^2 -
          > b^4 c^2 x^4 y^2 + 2 a^2 c^4 x^4 y^2 + 2 b^2 c^4 x^4 y^2 -
          > c^6 x^4 y^2 - a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 -
          > b^4 c^2 x^3 y^3 + c^6 x^3 y^3 - a^6 x^4 y z + 3 a^4 b^2 x^4 y z -
          > 3 a^2 b^4 x^4 y z + b^6 x^4 y z + 3 a^4 c^2 x^4 y z -
          > 2 a^2 b^2 c^2 x^4 y z - b^4 c^2 x^4 y z - 3 a^2 c^4 x^4 y z -
          > b^2 c^4 x^4 y z + c^6 x^4 y z + a^6 x^3 y^2 z -
          > 3 a^4 b^2 x^3 y^2 z + 3 a^2 b^4 x^3 y^2 z - b^6 x^3 y^2 z -
          > 2 a^4 c^2 x^3 y^2 z - 6 a^2 b^2 c^2 x^3 y^2 z +
          > 8 b^4 c^2 x^3 y^2 z + a^2 c^4 x^3 y^2 z - 7 b^2 c^4 x^3 y^2 z +
          > 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
          > b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
          > c^6 x^2 y^3 z - a^4 b^2 x^4 z^2 + 2 a^2 b^4 x^4 z^2 -
          > b^6 x^4 z^2 + 2 a^2 b^2 c^2 x^4 z^2 + 2 b^4 c^2 x^4 z^2 -
          > b^2 c^4 x^4 z^2 + a^6 x^3 y z^2 - 2 a^4 b^2 x^3 y z^2 +
          > a^2 b^4 x^3 y z^2 - 3 a^4 c^2 x^3 y z^2 -
          > 6 a^2 b^2 c^2 x^3 y z^2 - 7 b^4 c^2 x^3 y z^2 +
          > 3 a^2 c^4 x^3 y z^2 + 8 b^2 c^4 x^3 y z^2 - c^6 x^3 y z^2 +
          > a^4 b^2 x^2 y^2 z^2 - 2 a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 +
          > a^4 c^2 x^2 y^2 z^2 + 20 a^2 b^2 c^2 x^2 y^2 z^2 -
          > b^4 c^2 x^2 y^2 z^2 - 2 a^2 c^4 x^2 y^2 z^2 -
          > b^2 c^4 x^2 y^2 z^2 + c^6 x^2 y^2 z^2 - a^6 x y^3 z^2 +
          > 2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
          > 4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
          > b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 +
          > 5 a^4 b^2 x^2 y z^3 - 4 a^2 b^4 x^2 y z^3 - b^6 x^2 y z^3 -
          > 4 a^2 b^2 c^2 x^2 y z^3 + 2 b^4 c^2 x^2 y z^3 -
          > b^2 c^4 x^2 y z^3 - a^6 x y^2 z^3 - 4 a^4 b^2 x y^2 z^3 +
          > 5 a^2 b^4 x y^2 z^3 + 2 a^4 c^2 x y^2 z^3 -
          > 4 a^2 b^2 c^2 x y^2 z^3 - a^2 c^4 x y^2 z^3 + a^6 y^3 z^3 -
          > a^2 b^4 y^3 z^3 + 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3), (x -
          > z) (a^4 c^2 x^3 y^3 - 2 a^2 b^2 c^2 x^3 y^3 + b^4 c^2 x^3 y^3 -
          > c^6 x^3 y^3 + a^4 c^2 x^2 y^4 - 2 a^2 b^2 c^2 x^2 y^4 +
          > b^4 c^2 x^2 y^4 - 2 a^2 c^4 x^2 y^4 - 2 b^2 c^4 x^2 y^4 +
          > c^6 x^2 y^4 + a^4 c^2 x^3 y^2 z + 4 a^2 b^2 c^2 x^3 y^2 z -
          > 5 b^4 c^2 x^3 y^2 z - 2 a^2 c^4 x^3 y^2 z + 4 b^2 c^4 x^3 y^2 z +
          > c^6 x^3 y^2 z + a^6 x^2 y^3 z - 3 a^4 b^2 x^2 y^3 z +
          > 3 a^2 b^4 x^2 y^3 z - b^6 x^2 y^3 z - 8 a^4 c^2 x^2 y^3 z +
          > 6 a^2 b^2 c^2 x^2 y^3 z + 2 b^4 c^2 x^2 y^3 z +
          > 7 a^2 c^4 x^2 y^3 z - b^2 c^4 x^2 y^3 z - a^6 x y^4 z +
          > 3 a^4 b^2 x y^4 z - 3 a^2 b^4 x y^4 z + b^6 x y^4 z +
          > a^4 c^2 x y^4 z + 2 a^2 b^2 c^2 x y^4 z - 3 b^4 c^2 x y^4 z +
          > a^2 c^4 x y^4 z + 3 b^2 c^4 x y^4 z - c^6 x y^4 z +
          > a^4 b^2 x^3 y z^2 - 2 a^2 b^4 x^3 y z^2 + b^6 x^3 y z^2 +
          > 4 a^2 b^2 c^2 x^3 y z^2 + 4 b^4 c^2 x^3 y z^2 -
          > 5 b^2 c^4 x^3 y z^2 - a^6 x^2 y^2 z^2 + 2 a^4 b^2 x^2 y^2 z^2 -
          > a^2 b^4 x^2 y^2 z^2 + a^4 c^2 x^2 y^2 z^2 -
          > 20 a^2 b^2 c^2 x^2 y^2 z^2 - b^4 c^2 x^2 y^2 z^2 +
          > a^2 c^4 x^2 y^2 z^2 + 2 b^2 c^4 x^2 y^2 z^2 - c^6 x^2 y^2 z^2 -
          > a^4 b^2 x y^3 z^2 + 2 a^2 b^4 x y^3 z^2 - b^6 x y^3 z^2 +
          > 7 a^4 c^2 x y^3 z^2 + 6 a^2 b^2 c^2 x y^3 z^2 +
          > 3 b^4 c^2 x y^3 z^2 - 8 a^2 c^4 x y^3 z^2 - 3 b^2 c^4 x y^3 z^2 +
          > c^6 x y^3 z^2 + a^6 y^4 z^2 - 2 a^4 b^2 y^4 z^2 +
          > a^2 b^4 y^4 z^2 - 2 a^4 c^2 y^4 z^2 - 2 a^2 b^2 c^2 y^4 z^2 +
          > a^2 c^4 y^4 z^2 + a^4 b^2 x^3 z^3 - b^6 x^3 z^3 -
          > 2 a^2 b^2 c^2 x^3 z^3 + b^2 c^4 x^3 z^3 - 5 a^4 b^2 x^2 y z^3 +
          > 4 a^2 b^4 x^2 y z^3 + b^6 x^2 y z^3 + 4 a^2 b^2 c^2 x^2 y z^3 -
          > 2 b^4 c^2 x^2 y z^3 + b^2 c^4 x^2 y z^3 + a^6 x y^2 z^3 +
          > 4 a^4 b^2 x y^2 z^3 - 5 a^2 b^4 x y^2 z^3 - 2 a^4 c^2 x y^2 z^3 +
          > 4 a^2 b^2 c^2 x y^2 z^3 + a^2 c^4 x y^2 z^3 - a^6 y^3 z^3 +
          > a^2 b^4 y^3 z^3 - 2 a^2 b^2 c^2 y^3 z^3 + a^2 c^4 y^3 z^3), (x -
          > y) (-a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 - b^4 c^2 x^3 y^3 +
          > c^6 x^3 y^3 - a^4 c^2 x^3 y^2 z - 4 a^2 b^2 c^2 x^3 y^2 z +
          > 5 b^4 c^2 x^3 y^2 z + 2 a^2 c^4 x^3 y^2 z - 4 b^2 c^4 x^3 y^2 z -
          > c^6 x^3 y^2 z + 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
          > b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
          > c^6 x^2 y^3 z - a^4 b^2 x^3 y z^2 + 2 a^2 b^4 x^3 y z^2 -
          > b^6 x^3 y z^2 - 4 a^2 b^2 c^2 x^3 y z^2 - 4 b^4 c^2 x^3 y z^2 +
          > 5 b^2 c^4 x^3 y z^2 + a^6 x^2 y^2 z^2 - a^4 b^2 x^2 y^2 z^2 -
          > a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 - 2 a^4 c^2 x^2 y^2 z^2 +
          > 20 a^2 b^2 c^2 x^2 y^2 z^2 - 2 b^4 c^2 x^2 y^2 z^2 +
          > a^2 c^4 x^2 y^2 z^2 + b^2 c^4 x^2 y^2 z^2 - a^6 x y^3 z^2 +
          > 2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
          > 4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
          > b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 -
          > a^6 x^2 y z^3 + 8 a^4 b^2 x^2 y z^3 - 7 a^2 b^4 x^2 y z^3 +
          > 3 a^4 c^2 x^2 y z^3 - 6 a^2 b^2 c^2 x^2 y z^3 +
          > b^4 c^2 x^2 y z^3 - 3 a^2 c^4 x^2 y z^3 - 2 b^2 c^4 x^2 y z^3 +
          > c^6 x^2 y z^3 - 7 a^4 b^2 x y^2 z^3 + 8 a^2 b^4 x y^2 z^3 -
          > b^6 x y^2 z^3 + a^4 c^2 x y^2 z^3 - 6 a^2 b^2 c^2 x y^2 z^3 +
          > 3 b^4 c^2 x y^2 z^3 - 2 a^2 c^4 x y^2 z^3 - 3 b^2 c^4 x y^2 z^3 +
          > c^6 x y^2 z^3 + a^6 y^3 z^3 - a^2 b^4 y^3 z^3 +
          > 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3 - a^4 b^2 x^2 z^4 +
          > 2 a^2 b^4 x^2 z^4 - b^6 x^2 z^4 + 2 a^2 b^2 c^2 x^2 z^4 +
          > 2 b^4 c^2 x^2 z^4 - b^2 c^4 x^2 z^4 + a^6 x y z^4 -
          > a^4 b^2 x y z^4 - a^2 b^4 x y z^4 + b^6 x y z^4 -
          > 3 a^4 c^2 x y z^4 - 2 a^2 b^2 c^2 x y z^4 - 3 b^4 c^2 x y z^4 +
          > 3 a^2 c^4 x y z^4 + 3 b^2 c^4 x y z^4 - c^6 x y z^4 -
          > a^6 y^2 z^4 + 2 a^4 b^2 y^2 z^4 - a^2 b^4 y^2 z^4 +
          > 2 a^4 c^2 y^2 z^4 + 2 a^2 b^2 c^2 y^2 z^4 - a^2 c^4 y^2 z^4)}
          >
          > --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
          > >
          > > Let T = ABC be a triangle and L a line intersecting
          > > the sidelines BC,CA,AB of ABC at A', B', C', resp.
          > >
          > > Description of a transformation f of T in f(T):
          > >
          > > Let f(a) be the reflection of the perpendicular to BC at A'
          > > in the line L, and similarly f(b), f(c).
          > >
          > > Denote: f(T) = the triangle bounded by the lines f(a), f(b), f(c).
          > >
          > > The triangles T, f(T) are parallelogic and let x,y be the parallelogic centers.
          > >
          > > We have ff(T) = T and f(x) = y and f(y) = x
          > > [==> ff(x) = f(y) = x and ff(y) = f(x) = y]
          > >
          > > Which are the coordinates of x,y ?
          > >
          > > APH
          > >
          >
        • Antreas Hatzipolakis
          The points X, Y are on the circumcircles of T, f(T), resp. Right? We can prove or disprove that X is on the circumcircle of T = ABC by this way: Let p(a),
          Message 4 of 4 , Jan 26, 2013
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            The points X, Y are on the circumcircles of T, f(T), resp. Right?

            We can prove or disprove that X is on the circumcircle of T = ABC by this
            way:

            Let p(a), p(b), p(c) be the parallels through A,B,C, to f(a), f(b), f(c),
            resp.
            and let r(a), r(b), r(c) their respective reflections on the angle
            bisectors of T .

            If the angles (L, r(a)), (L, r(b)), (L, r(c)) are equal, then the lines
            r(a), r(b), r(c)
            are parallel and therefore the isogonal conjugate of their point of
            concurrence (on the line at infinity)
            is lying on the circumcircle of ABC.

            APH


            2013/1/26 Francisco Javier <garciacapitan@...>

            > **
            >
            >
            > If L is the trilinear polar of P=(x:y:z) and
            > X=parallelogic center center of T with respect f(T)
            > Y = parallelogic center center of f(T) with respect T
            >
            > we have known points for X:
            >
            > for P=X1, X=X953,
            > for P=X4, X=X477
            > for P=X6, X=X2698
            > for P=X7, X=X2724
            > for P=X8, X=X2734
            >
            > Coordinates of isotomic conjugate of X and coordinates of Y are below.
            >
            > Best regards,
            >
            > Francisco Javier.
            >
            > ----------------
            >
            > Coordinates of isotomic conjugate of X are
            >
            > {a^2 c^2 x^2 y^2 - b^2 c^2 x^2 y^2 - c^4 x^2 y^2 +
            > 4 b^2 c^2 x^2 y z - 2 a^2 c^2 x y^2 z - 2 b^2 c^2 x y^2 z +
            > 2 c^4 x y^2 z + a^2 b^2 x^2 z^2 - b^4 x^2 z^2 - b^2 c^2 x^2 z^2 -
            > 2 a^2 b^2 x y z^2 + 2 b^4 x y z^2 - 2 b^2 c^2 x y z^2 +
            > a^2 b^2 y^2 z^2 - b^4 y^2 z^2 + a^2 c^2 y^2 z^2 +
            > 2 b^2 c^2 y^2 z^2 - c^4 y^2 z^2, -a^2 c^2 x^2 y^2 +
            > b^2 c^2 x^2 y^2 - c^4 x^2 y^2 - 2 a^2 c^2 x^2 y z -
            > 2 b^2 c^2 x^2 y z + 2 c^4 x^2 y z + 4 a^2 c^2 x y^2 z -
            > a^4 x^2 z^2 + a^2 b^2 x^2 z^2 + 2 a^2 c^2 x^2 z^2 +
            > b^2 c^2 x^2 z^2 - c^4 x^2 z^2 + 2 a^4 x y z^2 - 2 a^2 b^2 x y z^2 -
            > 2 a^2 c^2 x y z^2 - a^4 y^2 z^2 + a^2 b^2 y^2 z^2 -
            > a^2 c^2 y^2 z^2, -a^4 x^2 y^2 + 2 a^2 b^2 x^2 y^2 - b^4 x^2 y^2 +
            > a^2 c^2 x^2 y^2 + b^2 c^2 x^2 y^2 - 2 a^2 b^2 x^2 y z +
            > 2 b^4 x^2 y z - 2 b^2 c^2 x^2 y z + 2 a^4 x y^2 z -
            > 2 a^2 b^2 x y^2 z - 2 a^2 c^2 x y^2 z - a^2 b^2 x^2 z^2 -
            > b^4 x^2 z^2 + b^2 c^2 x^2 z^2 + 4 a^2 b^2 x y z^2 - a^4 y^2 z^2 -
            > a^2 b^2 y^2 z^2 + a^2 c^2 y^2 z^2}
            >
            > Coordinates of Y are
            >
            > {(y - z) (-a^4 c^2 x^4 y^2 + 2 a^2 b^2 c^2 x^4 y^2 -
            > b^4 c^2 x^4 y^2 + 2 a^2 c^4 x^4 y^2 + 2 b^2 c^4 x^4 y^2 -
            > c^6 x^4 y^2 - a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 -
            > b^4 c^2 x^3 y^3 + c^6 x^3 y^3 - a^6 x^4 y z + 3 a^4 b^2 x^4 y z -
            > 3 a^2 b^4 x^4 y z + b^6 x^4 y z + 3 a^4 c^2 x^4 y z -
            > 2 a^2 b^2 c^2 x^4 y z - b^4 c^2 x^4 y z - 3 a^2 c^4 x^4 y z -
            > b^2 c^4 x^4 y z + c^6 x^4 y z + a^6 x^3 y^2 z -
            > 3 a^4 b^2 x^3 y^2 z + 3 a^2 b^4 x^3 y^2 z - b^6 x^3 y^2 z -
            > 2 a^4 c^2 x^3 y^2 z - 6 a^2 b^2 c^2 x^3 y^2 z +
            > 8 b^4 c^2 x^3 y^2 z + a^2 c^4 x^3 y^2 z - 7 b^2 c^4 x^3 y^2 z +
            > 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
            > b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
            > c^6 x^2 y^3 z - a^4 b^2 x^4 z^2 + 2 a^2 b^4 x^4 z^2 -
            > b^6 x^4 z^2 + 2 a^2 b^2 c^2 x^4 z^2 + 2 b^4 c^2 x^4 z^2 -
            > b^2 c^4 x^4 z^2 + a^6 x^3 y z^2 - 2 a^4 b^2 x^3 y z^2 +
            > a^2 b^4 x^3 y z^2 - 3 a^4 c^2 x^3 y z^2 -
            > 6 a^2 b^2 c^2 x^3 y z^2 - 7 b^4 c^2 x^3 y z^2 +
            > 3 a^2 c^4 x^3 y z^2 + 8 b^2 c^4 x^3 y z^2 - c^6 x^3 y z^2 +
            > a^4 b^2 x^2 y^2 z^2 - 2 a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 +
            > a^4 c^2 x^2 y^2 z^2 + 20 a^2 b^2 c^2 x^2 y^2 z^2 -
            > b^4 c^2 x^2 y^2 z^2 - 2 a^2 c^4 x^2 y^2 z^2 -
            > b^2 c^4 x^2 y^2 z^2 + c^6 x^2 y^2 z^2 - a^6 x y^3 z^2 +
            > 2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
            > 4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
            > b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 +
            > 5 a^4 b^2 x^2 y z^3 - 4 a^2 b^4 x^2 y z^3 - b^6 x^2 y z^3 -
            > 4 a^2 b^2 c^2 x^2 y z^3 + 2 b^4 c^2 x^2 y z^3 -
            > b^2 c^4 x^2 y z^3 - a^6 x y^2 z^3 - 4 a^4 b^2 x y^2 z^3 +
            > 5 a^2 b^4 x y^2 z^3 + 2 a^4 c^2 x y^2 z^3 -
            > 4 a^2 b^2 c^2 x y^2 z^3 - a^2 c^4 x y^2 z^3 + a^6 y^3 z^3 -
            > a^2 b^4 y^3 z^3 + 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3), (x -
            > z) (a^4 c^2 x^3 y^3 - 2 a^2 b^2 c^2 x^3 y^3 + b^4 c^2 x^3 y^3 -
            > c^6 x^3 y^3 + a^4 c^2 x^2 y^4 - 2 a^2 b^2 c^2 x^2 y^4 +
            > b^4 c^2 x^2 y^4 - 2 a^2 c^4 x^2 y^4 - 2 b^2 c^4 x^2 y^4 +
            > c^6 x^2 y^4 + a^4 c^2 x^3 y^2 z + 4 a^2 b^2 c^2 x^3 y^2 z -
            > 5 b^4 c^2 x^3 y^2 z - 2 a^2 c^4 x^3 y^2 z + 4 b^2 c^4 x^3 y^2 z +
            > c^6 x^3 y^2 z + a^6 x^2 y^3 z - 3 a^4 b^2 x^2 y^3 z +
            > 3 a^2 b^4 x^2 y^3 z - b^6 x^2 y^3 z - 8 a^4 c^2 x^2 y^3 z +
            > 6 a^2 b^2 c^2 x^2 y^3 z + 2 b^4 c^2 x^2 y^3 z +
            > 7 a^2 c^4 x^2 y^3 z - b^2 c^4 x^2 y^3 z - a^6 x y^4 z +
            > 3 a^4 b^2 x y^4 z - 3 a^2 b^4 x y^4 z + b^6 x y^4 z +
            > a^4 c^2 x y^4 z + 2 a^2 b^2 c^2 x y^4 z - 3 b^4 c^2 x y^4 z +
            > a^2 c^4 x y^4 z + 3 b^2 c^4 x y^4 z - c^6 x y^4 z +
            > a^4 b^2 x^3 y z^2 - 2 a^2 b^4 x^3 y z^2 + b^6 x^3 y z^2 +
            > 4 a^2 b^2 c^2 x^3 y z^2 + 4 b^4 c^2 x^3 y z^2 -
            > 5 b^2 c^4 x^3 y z^2 - a^6 x^2 y^2 z^2 + 2 a^4 b^2 x^2 y^2 z^2 -
            > a^2 b^4 x^2 y^2 z^2 + a^4 c^2 x^2 y^2 z^2 -
            > 20 a^2 b^2 c^2 x^2 y^2 z^2 - b^4 c^2 x^2 y^2 z^2 +
            > a^2 c^4 x^2 y^2 z^2 + 2 b^2 c^4 x^2 y^2 z^2 - c^6 x^2 y^2 z^2 -
            > a^4 b^2 x y^3 z^2 + 2 a^2 b^4 x y^3 z^2 - b^6 x y^3 z^2 +
            > 7 a^4 c^2 x y^3 z^2 + 6 a^2 b^2 c^2 x y^3 z^2 +
            > 3 b^4 c^2 x y^3 z^2 - 8 a^2 c^4 x y^3 z^2 - 3 b^2 c^4 x y^3 z^2 +
            > c^6 x y^3 z^2 + a^6 y^4 z^2 - 2 a^4 b^2 y^4 z^2 +
            > a^2 b^4 y^4 z^2 - 2 a^4 c^2 y^4 z^2 - 2 a^2 b^2 c^2 y^4 z^2 +
            > a^2 c^4 y^4 z^2 + a^4 b^2 x^3 z^3 - b^6 x^3 z^3 -
            > 2 a^2 b^2 c^2 x^3 z^3 + b^2 c^4 x^3 z^3 - 5 a^4 b^2 x^2 y z^3 +
            > 4 a^2 b^4 x^2 y z^3 + b^6 x^2 y z^3 + 4 a^2 b^2 c^2 x^2 y z^3 -
            > 2 b^4 c^2 x^2 y z^3 + b^2 c^4 x^2 y z^3 + a^6 x y^2 z^3 +
            > 4 a^4 b^2 x y^2 z^3 - 5 a^2 b^4 x y^2 z^3 - 2 a^4 c^2 x y^2 z^3 +
            > 4 a^2 b^2 c^2 x y^2 z^3 + a^2 c^4 x y^2 z^3 - a^6 y^3 z^3 +
            > a^2 b^4 y^3 z^3 - 2 a^2 b^2 c^2 y^3 z^3 + a^2 c^4 y^3 z^3), (x -
            > y) (-a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 - b^4 c^2 x^3 y^3 +
            > c^6 x^3 y^3 - a^4 c^2 x^3 y^2 z - 4 a^2 b^2 c^2 x^3 y^2 z +
            > 5 b^4 c^2 x^3 y^2 z + 2 a^2 c^4 x^3 y^2 z - 4 b^2 c^4 x^3 y^2 z -
            > c^6 x^3 y^2 z + 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
            > b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
            > c^6 x^2 y^3 z - a^4 b^2 x^3 y z^2 + 2 a^2 b^4 x^3 y z^2 -
            > b^6 x^3 y z^2 - 4 a^2 b^2 c^2 x^3 y z^2 - 4 b^4 c^2 x^3 y z^2 +
            > 5 b^2 c^4 x^3 y z^2 + a^6 x^2 y^2 z^2 - a^4 b^2 x^2 y^2 z^2 -
            > a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 - 2 a^4 c^2 x^2 y^2 z^2 +
            > 20 a^2 b^2 c^2 x^2 y^2 z^2 - 2 b^4 c^2 x^2 y^2 z^2 +
            > a^2 c^4 x^2 y^2 z^2 + b^2 c^4 x^2 y^2 z^2 - a^6 x y^3 z^2 +
            > 2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
            > 4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
            > b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 -
            > a^6 x^2 y z^3 + 8 a^4 b^2 x^2 y z^3 - 7 a^2 b^4 x^2 y z^3 +
            > 3 a^4 c^2 x^2 y z^3 - 6 a^2 b^2 c^2 x^2 y z^3 +
            > b^4 c^2 x^2 y z^3 - 3 a^2 c^4 x^2 y z^3 - 2 b^2 c^4 x^2 y z^3 +
            > c^6 x^2 y z^3 - 7 a^4 b^2 x y^2 z^3 + 8 a^2 b^4 x y^2 z^3 -
            > b^6 x y^2 z^3 + a^4 c^2 x y^2 z^3 - 6 a^2 b^2 c^2 x y^2 z^3 +
            > 3 b^4 c^2 x y^2 z^3 - 2 a^2 c^4 x y^2 z^3 - 3 b^2 c^4 x y^2 z^3 +
            > c^6 x y^2 z^3 + a^6 y^3 z^3 - a^2 b^4 y^3 z^3 +
            > 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3 - a^4 b^2 x^2 z^4 +
            > 2 a^2 b^4 x^2 z^4 - b^6 x^2 z^4 + 2 a^2 b^2 c^2 x^2 z^4 +
            > 2 b^4 c^2 x^2 z^4 - b^2 c^4 x^2 z^4 + a^6 x y z^4 -
            > a^4 b^2 x y z^4 - a^2 b^4 x y z^4 + b^6 x y z^4 -
            > 3 a^4 c^2 x y z^4 - 2 a^2 b^2 c^2 x y z^4 - 3 b^4 c^2 x y z^4 +
            > 3 a^2 c^4 x y z^4 + 3 b^2 c^4 x y z^4 - c^6 x y z^4 -
            > a^6 y^2 z^4 + 2 a^4 b^2 y^2 z^4 - a^2 b^4 y^2 z^4 +
            > 2 a^4 c^2 y^2 z^4 + 2 a^2 b^2 c^2 y^2 z^4 - a^2 c^4 y^2 z^4)}
            >
            >
            > --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
            > >
            > > Let T = ABC be a triangle and L a line intersecting
            > > the sidelines BC,CA,AB of ABC at A', B', C', resp.
            > >
            > > Description of a transformation f of T in f(T):
            > >
            > > Let f(a) be the reflection of the perpendicular to BC at A'
            > > in the line L, and similarly f(b), f(c).
            > >
            > > Denote: f(T) = the triangle bounded by the lines f(a), f(b), f(c).
            > >
            > > The triangles T, f(T) are parallelogic and let x,y be the parallelogic
            > centers.
            > >
            > > We have ff(T) = T and f(x) = y and f(y) = x
            > > [==> ff(x) = f(y) = x and ff(y) = f(x) = y]
            > >
            > > Which are the coordinates of x,y ?
            > >
            > > APH
            > >
            >
            > _
            >


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