Loading ...
Sorry, an error occurred while loading the content.

Re: Parallelogic triangles [corrected]

Expand Messages
  • Francisco Javier
    Sorry, I meant Y = parallelogic center center of f(T) with respect T
    Message 1 of 4 , Jan 25, 2013
    • 0 Attachment
      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 2 of 4 , Jan 26, 2013
      • 0 Attachment
        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
        > >
        >
        > _
        >


        [Non-text portions of this message have been removed]
      Your message has been successfully submitted and would be delivered to recipients shortly.