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The Fregier point

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  • Nikolaos Dergiades
    Dear friends, it is known the Fregier s theorem: If AXY is a right agnled triangle at A and is inscribed in a conic Co with A constant and XY variable then XY
    Message 1 of 7 , Sep 6, 2012
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      Dear friends,
      it is known the Fregier's theorem:
      If AXY is a right agnled triangle at A
      and is inscribed in a conic Co with A constant and XY variable then
      XY passes through a constant point P.

      We call this point P the Fregier point Fr(A) of A relative
      to this conic Co.
      If A moves on Co then its Fregier point moves on another conic
      that we call the Fregier conic of Co.

      Now if Co is a circumconic of triangle ABC with equation in barycentrics
      pyz + qzx + rxy = 0 then the Fregier conic of this circumconic is a circle
      if the point (p : q : r) lies on the orthic axis of ABC.

      If the circumconic Co is the Steiner circumellipse and
      P = (x : y : z) in bars
      is a point on it then the Fregier point of P is
      Fr(P)=(xSA+ySC+zSB : xSC+ySB+zSA : xSB+ySA+zSC)
      and the Fregier conic of the Co is an ellipse homothetic to Co
      with center G and ratio of homothecy
      2.sqrt[a^4+b^4+c^4-(bc)^2-(ca)^2-(ab)^2] / (aa+bb+cc).
      I think that from the 23 known in ETC points of the
      Steiner circumellipse only two have Fregier points on ETC that is:
      Fr( X(99) ) = X(69)
      Fr( X(671) ) = X(1992).

      Best regards
      Nikos Dergiades
    • Francois Rideau
      Dear Nikos I have always thought that the Fregier conic of a conic Gamma was (indirectly) similar to Gamma. Moreover circumconics with perspectors on orthic
      Message 2 of 7 , Sep 6, 2012
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        Dear Nikos
        I have always thought that the Fregier conic of a conic Gamma was
        (indirectly) similar to Gamma.
        Moreover circumconics with perspectors on orthic axis are rectangular
        hyperbola and so their Fregier conics are also rectangular hyperbola.
        Friendly Francois

        On Thu, Sep 6, 2012 at 8:24 PM, Nikolaos Dergiades <ndergiades@...>wrote:

        > **
        >
        >
        > Dear friends,
        > it is known the Fregier's theorem:
        > If AXY is a right agnled triangle at A
        > and is inscribed in a conic Co with A constant and XY variable then
        > XY passes through a constant point P.
        >
        > We call this point P the Fregier point Fr(A) of A relative
        > to this conic Co.
        > If A moves on Co then its Fregier point moves on another conic
        > that we call the Fregier conic of Co.
        >
        > Now if Co is a circumconic of triangle ABC with equation in barycentrics
        > pyz + qzx + rxy = 0 then the Fregier conic of this circumconic is a circle
        > if the point (p : q : r) lies on the orthic axis of ABC.
        >
        > If the circumconic Co is the Steiner circumellipse and
        > P = (x : y : z) in bars
        > is a point on it then the Fregier point of P is
        > Fr(P)=(xSA+ySC+zSB : xSC+ySB+zSA : xSB+ySA+zSC)
        > and the Fregier conic of the Co is an ellipse homothetic to Co
        > with center G and ratio of homothecy
        > 2.sqrt[a^4+b^4+c^4-(bc)^2-(ca)^2-(ab)^2] / (aa+bb+cc).
        > I think that from the 23 known in ETC points of the
        > Steiner circumellipse only two have Fregier points on ETC that is:
        > Fr( X(99) ) = X(69)
        > Fr( X(671) ) = X(1992).
        >
        > Best regards
        > Nikos Dergiades
        >
        >
        >


        [Non-text portions of this message have been removed]
      • Nikolaos Dergiades
        Dear Francois I didn t know that are indirectly similar. So maybe I have made a mistake in my calculations. So synthetic knowledge is stronger than analytic.
        Message 3 of 7 , Sep 6, 2012
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          Dear Francois
          I didn't know that are indirectly similar.
          So maybe I have made a mistake in my
          calculations. So synthetic knowledge
          is stronger than analytic.
          Thank you very much.
          Best regards
          Nikos Dergiades


          > Dear Nikos
          > I have always thought that the Fregier conic of a conic
          > Gamma was
          > (indirectly) similar to Gamma.
          > Moreover circumconics with perspectors on orthic axis are
          > rectangular
          > hyperbola  and so their Fregier conics are also
          > rectangular hyperbola.
          > Friendly Francois
          >
          > On Thu, Sep 6, 2012 at 8:24 PM, Nikolaos Dergiades <ndergiades@...>wrote:
          >
          > > **
          > >
          > >
          > > Dear friends,
          > > it is known the Fregier's theorem:
          > > If AXY is a right agnled triangle at A
          > > and is inscribed in a conic Co with A constant and XY
          > variable then
          > > XY passes through a constant point P.
          > >
          > > We call this point P the Fregier point Fr(A) of A
          > relative
          > > to this conic Co.
          > > If A moves on Co then its Fregier point moves on
          > another conic
          > > that we call the Fregier conic of Co.
          > >
          > > Now if Co is a circumconic of triangle ABC with
          > equation in barycentrics
          > > pyz + qzx + rxy = 0 then the Fregier conic of this
          > circumconic is a circle
          > > if the point (p : q : r) lies on the orthic axis of
          > ABC.
          > >
          > > If the circumconic Co is the Steiner circumellipse and
          > > P = (x : y : z) in bars
          > > is a point on it then the Fregier point of P is
          > > Fr(P)=(xSA+ySC+zSB : xSC+ySB+zSA : xSB+ySA+zSC)
          > > and the Fregier conic of the Co is an ellipse
          > homothetic to Co
          > > with center G and ratio of homothecy
          > > 2.sqrt[a^4+b^4+c^4-(bc)^2-(ca)^2-(ab)^2] / (aa+bb+cc).
          > > I think that from the 23 known in ETC points of the
          > > Steiner circumellipse only two have Fregier points on
          > ETC that is:
          > > Fr( X(99) ) = X(69)
          > > Fr( X(671) ) = X(1992).
          > >
          > > Best regards
          > > Nikos Dergiades
          > >
          > > 
          > >
          >
          >
          > [Non-text portions of this message have been removed]
          >
          >
          >
          > ------------------------------------
          >
          > Yahoo! Groups Links
          >
          >
          >     Hyacinthos-fullfeatured@yahoogroups.com
          >
          >
        • Nikolaos Dergiades
          Dear Francois, I wrote that if the perspector of a circumconic lies on the Orthic axis then its Fregier conic satisfies the condition to be a circle. I did not
          Message 4 of 7 , Sep 7, 2012
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            Dear Francois,
            I wrote that if the perspector of a circumconic
            lies on the Orthic axis then its Fregier conic
            satisfies the condition to be a circle.
            I did not observed that there is more.
            The Fregier conic satisfies and other conditions
            and I think that is the line at infinity.

            If my sketch is correct then if the perspector
            of a circumconic lies on the orthic axis or
            if the center of the circumconic lies on the NPC
            then the conic is a rectangular hyperbola Co
            and the hypotenuse XY of the right angled triangle
            AXY inscribed in Co is parallel to the normal
            of Co at A. Hence the Fregier conic of Co is the line
            at infinity.
            Best regards
            Nikos Dergiades




            > Dear Nikos
            > I have always thought that the Fregier conic of a conic
            > Gamma was
            > (indirectly) similar to Gamma.
            > Moreover circumconics with perspectors on orthic axis are
            > rectangular
            > hyperbola  and so their Fregier conics are also
            > rectangular hyperbola.
            > Friendly Francois
            >
            > On Thu, Sep 6, 2012 at 8:24 PM, Nikolaos Dergiades <ndergiades@...>wrote:
            >
            > > **
            > >
            > >
            > > Dear friends,
            > > it is known the Fregier's theorem:
            > > If AXY is a right agnled triangle at A
            > > and is inscribed in a conic Co with A constant and XY
            > variable then
            > > XY passes through a constant point P.
            > >
            > > We call this point P the Fregier point Fr(A) of A
            > relative
            > > to this conic Co.
            > > If A moves on Co then its Fregier point moves on
            > another conic
            > > that we call the Fregier conic of Co.
            > >
            > > Now if Co is a circumconic of triangle ABC with
            > equation in barycentrics
            > > pyz + qzx + rxy = 0 then the Fregier conic of this
            > circumconic is a circle
            > > if the point (p : q : r) lies on the orthic axis of
            > ABC.
            > >
            > > If the circumconic Co is the Steiner circumellipse and
            > > P = (x : y : z) in bars
            > > is a point on it then the Fregier point of P is
            > > Fr(P)=(xSA+ySC+zSB : xSC+ySB+zSA : xSB+ySA+zSC)
            > > and the Fregier conic of the Co is an ellipse
            > homothetic to Co
            > > with center G and ratio of homothecy
            > > 2.sqrt[a^4+b^4+c^4-(bc)^2-(ca)^2-(ab)^2] / (aa+bb+cc).
            > > I think that from the 23 known in ETC points of the
            > > Steiner circumellipse only two have Fregier points on
            > ETC that is:
            > > Fr( X(99) ) = X(69)
            > > Fr( X(671) ) = X(1992).
            > >
            > > Best regards
            > > Nikos Dergiades
            > >
            > > 
            > >
            >
            >
            > [Non-text portions of this message have been removed]
            >
            >
            >
            > ------------------------------------
            >
            > Yahoo! Groups Links
            >
            >
            >     Hyacinthos-fullfeatured@yahoogroups.com
            >
            >
          • Francois Rideau
            Dear Nikos The best to get the Fregier conic is to make calculations on a reduced form, for example: x²/a²+y²/b²-1 = 0 for an ellipse or y² - 2px = 0 for
            Message 5 of 7 , Sep 8, 2012
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              Dear Nikos
              The best to get the Fregier conic is to make calculations on a reduced
              form, for example: x²/a²+y²/b²-1 = 0 for an ellipse or y² - 2px = 0 for a
              parabola or xy = k for a hyperbola.
              If you want a more synthetic way, look at what happens when the sides of
              the right angle through point M on the conic are parallel to the axis of
              this conic.
              Friendly
              Francois

              On Fri, Sep 7, 2012 at 11:40 AM, Nikolaos Dergiades <ndergiades@...>wrote:

              > Dear Francois,
              > I wrote that if the perspector of a circumconic
              > lies on the Orthic axis then its Fregier conic
              > satisfies the condition to be a circle.
              > I did not observed that there is more.
              > The Fregier conic satisfies and other conditions
              > and I think that is the line at infinity.
              >
              > If my sketch is correct then if the perspector
              > of a circumconic lies on the orthic axis or
              > if the center of the circumconic lies on the NPC
              > then the conic is a rectangular hyperbola Co
              > and the hypotenuse XY of the right angled triangle
              > AXY inscribed in Co is parallel to the normal
              > of Co at A. Hence the Fregier conic of Co is the line
              > at infinity.
              > Best regards
              > Nikos Dergiades
              >
              >
              >
              >
              > > Dear Nikos
              > > I have always thought that the Fregier conic of a conic
              > > Gamma was
              > > (indirectly) similar to Gamma.
              > > Moreover circumconics with perspectors on orthic axis are
              > > rectangular
              > > hyperbola and so their Fregier conics are also
              > > rectangular hyperbola.
              > > Friendly Francois
              > >
              > > On Thu, Sep 6, 2012 at 8:24 PM, Nikolaos Dergiades <ndergiades@...
              > >wrote:
              > >
              > > > **
              > > >
              > > >
              > > > Dear friends,
              > > > it is known the Fregier's theorem:
              > > > If AXY is a right agnled triangle at A
              > > > and is inscribed in a conic Co with A constant and XY
              > > variable then
              > > > XY passes through a constant point P.
              > > >
              > > > We call this point P the Fregier point Fr(A) of A
              > > relative
              > > > to this conic Co.
              > > > If A moves on Co then its Fregier point moves on
              > > another conic
              > > > that we call the Fregier conic of Co.
              > > >
              > > > Now if Co is a circumconic of triangle ABC with
              > > equation in barycentrics
              > > > pyz + qzx + rxy = 0 then the Fregier conic of this
              > > circumconic is a circle
              > > > if the point (p : q : r) lies on the orthic axis of
              > > ABC.
              > > >
              > > > If the circumconic Co is the Steiner circumellipse and
              > > > P = (x : y : z) in bars
              > > > is a point on it then the Fregier point of P is
              > > > Fr(P)=(xSA+ySC+zSB : xSC+ySB+zSA : xSB+ySA+zSC)
              > > > and the Fregier conic of the Co is an ellipse
              > > homothetic to Co
              > > > with center G and ratio of homothecy
              > > > 2.sqrt[a^4+b^4+c^4-(bc)^2-(ca)^2-(ab)^2] / (aa+bb+cc).
              > > > I think that from the 23 known in ETC points of the
              > > > Steiner circumellipse only two have Fregier points on
              > > ETC that is:
              > > > Fr( X(99) ) = X(69)
              > > > Fr( X(671) ) = X(1992).
              > > >
              > > > Best regards
              > > > Nikos Dergiades
              > > >
              > > >
              > > >
              > >
              > >
              > > [Non-text portions of this message have been removed]
              > >
              > >
              > >
              > > ------------------------------------
              > >
              > > Yahoo! Groups Links
              > >
              > >
              > > Hyacinthos-fullfeatured@yahoogroups.com
              > >
              > >
              >
              >
              > ------------------------------------
              >
              > Yahoo! Groups Links
              >
              >
              >
              >


              [Non-text portions of this message have been removed]
            • Francois Rideau
              Another remark: The Fregier theorem is still true in any dimension under some general conditions. Let S be a quadric in a n-dimensional euclidian space and M
              Message 6 of 7 , Sep 8, 2012
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                Another remark:
                The Fregier theorem is still true in any dimension under some general
                conditions.
                Let S be a quadric in a n-dimensional euclidian space and M be some fixed
                point on S
                Let (Mx_1, Mx_2, ..., Mx_n) n lines through M mutually orthogonal.
                They meet again S in n other points A_1, A_2,..., A_n.
                Then the hyperplane on A_1, A_2, ..., A_n is on some fixed point H
                belonging to the normal at S on h.
                The locus of the Fregier point H is another quadric similar to S.
                Friendly
                Francois
                PS
                Of course, some calculations is needed for its proof!


                On Sat, Sep 8, 2012 at 12:27 PM, Francois Rideau
                <francois.rideau@...>wrote:

                > Dear Nikos
                > The best to get the Fregier conic is to make calculations on a reduced
                > form, for example: x²/a²+y²/b²-1 = 0 for an ellipse or y² - 2px = 0 for a
                > parabola or xy = k for a hyperbola.
                > If you want a more synthetic way, look at what happens when the sides of
                > the right angle through point M on the conic are parallel to the axis of
                > this conic.
                > Friendly
                > Francois
                >
                >
                > On Fri, Sep 7, 2012 at 11:40 AM, Nikolaos Dergiades <ndergiades@...>wrote:
                >
                >> Dear Francois,
                >> I wrote that if the perspector of a circumconic
                >> lies on the Orthic axis then its Fregier conic
                >> satisfies the condition to be a circle.
                >> I did not observed that there is more.
                >> The Fregier conic satisfies and other conditions
                >> and I think that is the line at infinity.
                >>
                >> If my sketch is correct then if the perspector
                >> of a circumconic lies on the orthic axis or
                >> if the center of the circumconic lies on the NPC
                >> then the conic is a rectangular hyperbola Co
                >> and the hypotenuse XY of the right angled triangle
                >> AXY inscribed in Co is parallel to the normal
                >> of Co at A. Hence the Fregier conic of Co is the line
                >> at infinity.
                >> Best regards
                >> Nikos Dergiades
                >>
                >>
                >>
                >>
                >> > Dear Nikos
                >> > I have always thought that the Fregier conic of a conic
                >> > Gamma was
                >> > (indirectly) similar to Gamma.
                >> > Moreover circumconics with perspectors on orthic axis are
                >> > rectangular
                >> > hyperbola and so their Fregier conics are also
                >> > rectangular hyperbola.
                >> > Friendly Francois
                >> >
                >> > On Thu, Sep 6, 2012 at 8:24 PM, Nikolaos Dergiades <ndergiades@...
                >> >wrote:
                >> >
                >> > > **
                >> > >
                >> > >
                >> > > Dear friends,
                >> > > it is known the Fregier's theorem:
                >> > > If AXY is a right agnled triangle at A
                >> > > and is inscribed in a conic Co with A constant and XY
                >> > variable then
                >> > > XY passes through a constant point P.
                >> > >
                >> > > We call this point P the Fregier point Fr(A) of A
                >> > relative
                >> > > to this conic Co.
                >> > > If A moves on Co then its Fregier point moves on
                >> > another conic
                >> > > that we call the Fregier conic of Co.
                >> > >
                >> > > Now if Co is a circumconic of triangle ABC with
                >> > equation in barycentrics
                >> > > pyz + qzx + rxy = 0 then the Fregier conic of this
                >> > circumconic is a circle
                >> > > if the point (p : q : r) lies on the orthic axis of
                >> > ABC.
                >> > >
                >> > > If the circumconic Co is the Steiner circumellipse and
                >> > > P = (x : y : z) in bars
                >> > > is a point on it then the Fregier point of P is
                >> > > Fr(P)=(xSA+ySC+zSB : xSC+ySB+zSA : xSB+ySA+zSC)
                >> > > and the Fregier conic of the Co is an ellipse
                >> > homothetic to Co
                >> > > with center G and ratio of homothecy
                >> > > 2.sqrt[a^4+b^4+c^4-(bc)^2-(ca)^2-(ab)^2] / (aa+bb+cc).
                >> > > I think that from the 23 known in ETC points of the
                >> > > Steiner circumellipse only two have Fregier points on
                >> > ETC that is:
                >> > > Fr( X(99) ) = X(69)
                >> > > Fr( X(671) ) = X(1992).
                >> > >
                >> > > Best regards
                >> > > Nikos Dergiades
                >> > >
                >> > >
                >> > >
                >> >
                >> >
                >> > [Non-text portions of this message have been removed]
                >> >
                >> >
                >> >
                >> > ------------------------------------
                >> >
                >> > Yahoo! Groups Links
                >> >
                >> >
                >> > Hyacinthos-fullfeatured@yahoogroups.com
                >> >
                >> >
                >>
                >>
                >> ------------------------------------
                >>
                >> Yahoo! Groups Links
                >>
                >>
                >>
                >>
                >


                [Non-text portions of this message have been removed]
              • Francois Rideau
                Of course, you must read: Let S be a quadric in a n-dimensional euclidian space and M be some fixed point on S Let (Mx_1, Mx_2, ..., Mx_n) n lines through M
                Message 7 of 7 , Sep 8, 2012
                • 0 Attachment
                  Of course, you must read:
                  Let S be a quadric in a n-dimensional euclidian space and M be some fixed
                  point on S
                  Let (Mx_1, Mx_2, ..., Mx_n) n lines through M mutually orthogonal.
                  They meet again S in n other points A_1, A_2,..., A_n.
                  Then the hyperplane on A_1, A_2, ..., A_n is on some fixed point H
                  belonging to the normal at S on *M*.
                  The locus of the Fregier point H is another quadric similar to S.
                  Friendly


                  On Sat, Sep 8, 2012 at 12:38 PM, Francois Rideau
                  <francois.rideau@...>wrote:

                  > Another remark:
                  > The Fregier theorem is still true in any dimension under some general
                  > conditions.
                  > Let S be a quadric in a n-dimensional euclidian space and M be some fixed
                  > point on S
                  > Let (Mx_1, Mx_2, ..., Mx_n) n lines through M mutually orthogonal.
                  > They meet again S in n other points A_1, A_2,..., A_n.
                  > Then the hyperplane on A_1, A_2, ..., A_n is on some fixed point H
                  > belonging to the normal at S on h.
                  > The locus of the Fregier point H is another quadric similar to S.
                  > Friendly
                  > Francois
                  > PS
                  > Of course, some calculations is needed for its proof!
                  >
                  >
                  >
                  > On Sat, Sep 8, 2012 at 12:27 PM, Francois Rideau <
                  > francois.rideau@...> wrote:
                  >
                  >> Dear Nikos
                  >> The best to get the Fregier conic is to make calculations on a reduced
                  >> form, for example: x²/a²+y²/b²-1 = 0 for an ellipse or y² - 2px = 0 for a
                  >> parabola or xy = k for a hyperbola.
                  >> If you want a more synthetic way, look at what happens when the sides of
                  >> the right angle through point M on the conic are parallel to the axis of
                  >> this conic.
                  >> Friendly
                  >> Francois
                  >>
                  >>
                  >> On Fri, Sep 7, 2012 at 11:40 AM, Nikolaos Dergiades <ndergiades@...>wrote:
                  >>
                  >>> Dear Francois,
                  >>> I wrote that if the perspector of a circumconic
                  >>> lies on the Orthic axis then its Fregier conic
                  >>> satisfies the condition to be a circle.
                  >>> I did not observed that there is more.
                  >>> The Fregier conic satisfies and other conditions
                  >>> and I think that is the line at infinity.
                  >>>
                  >>> If my sketch is correct then if the perspector
                  >>> of a circumconic lies on the orthic axis or
                  >>> if the center of the circumconic lies on the NPC
                  >>> then the conic is a rectangular hyperbola Co
                  >>> and the hypotenuse XY of the right angled triangle
                  >>> AXY inscribed in Co is parallel to the normal
                  >>> of Co at A. Hence the Fregier conic of Co is the line
                  >>> at infinity.
                  >>> Best regards
                  >>> Nikos Dergiades
                  >>>
                  >>>
                  >>>
                  >>>
                  >>> > Dear Nikos
                  >>> > I have always thought that the Fregier conic of a conic
                  >>> > Gamma was
                  >>> > (indirectly) similar to Gamma.
                  >>> > Moreover circumconics with perspectors on orthic axis are
                  >>> > rectangular
                  >>> > hyperbola and so their Fregier conics are also
                  >>> > rectangular hyperbola.
                  >>> > Friendly Francois
                  >>> >
                  >>> > On Thu, Sep 6, 2012 at 8:24 PM, Nikolaos Dergiades <
                  >>> ndergiades@...>wrote:
                  >>> >
                  >>> > > **
                  >>> > >
                  >>> > >
                  >>> > > Dear friends,
                  >>> > > it is known the Fregier's theorem:
                  >>> > > If AXY is a right agnled triangle at A
                  >>> > > and is inscribed in a conic Co with A constant and XY
                  >>> > variable then
                  >>> > > XY passes through a constant point P.
                  >>> > >
                  >>> > > We call this point P the Fregier point Fr(A) of A
                  >>> > relative
                  >>> > > to this conic Co.
                  >>> > > If A moves on Co then its Fregier point moves on
                  >>> > another conic
                  >>> > > that we call the Fregier conic of Co.
                  >>> > >
                  >>> > > Now if Co is a circumconic of triangle ABC with
                  >>> > equation in barycentrics
                  >>> > > pyz + qzx + rxy = 0 then the Fregier conic of this
                  >>> > circumconic is a circle
                  >>> > > if the point (p : q : r) lies on the orthic axis of
                  >>> > ABC.
                  >>> > >
                  >>> > > If the circumconic Co is the Steiner circumellipse and
                  >>> > > P = (x : y : z) in bars
                  >>> > > is a point on it then the Fregier point of P is
                  >>> > > Fr(P)=(xSA+ySC+zSB : xSC+ySB+zSA : xSB+ySA+zSC)
                  >>> > > and the Fregier conic of the Co is an ellipse
                  >>> > homothetic to Co
                  >>> > > with center G and ratio of homothecy
                  >>> > > 2.sqrt[a^4+b^4+c^4-(bc)^2-(ca)^2-(ab)^2] / (aa+bb+cc).
                  >>> > > I think that from the 23 known in ETC points of the
                  >>> > > Steiner circumellipse only two have Fregier points on
                  >>> > ETC that is:
                  >>> > > Fr( X(99) ) = X(69)
                  >>> > > Fr( X(671) ) = X(1992).
                  >>> > >
                  >>> > > Best regards
                  >>> > > Nikos Dergiades
                  >>> > >
                  >>> > >
                  >>> > >
                  >>> >
                  >>> >
                  >>> > [Non-text portions of this message have been removed]
                  >>> >
                  >>> >
                  >>> >
                  >>> > ------------------------------------
                  >>> >
                  >>> > Yahoo! Groups Links
                  >>> >
                  >>> >
                  >>> > Hyacinthos-fullfeatured@yahoogroups.com
                  >>> >
                  >>> >
                  >>>
                  >>>
                  >>> ------------------------------------
                  >>>
                  >>> Yahoo! Groups Links
                  >>>
                  >>>
                  >>>
                  >>>
                  >>
                  >


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