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envelope (point)

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  • Antreas Hatzipolakis
    Let ABC be a point, L a line and A ,B ,C the orth. projections of A,B,C in L, resp. Let P be a point on L and Pa,Pb,Pc its reflections in AA ,BB ,CC , resp.
    Message 1 of 3 , Feb 27, 2013
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      Let ABC be a point, L a line and A',B',C' the orth. projections
      of A,B,C in L, resp.
      Let P be a point on L and Pa,Pb,Pc its reflections in AA',BB',CC', resp.
      The perpendiculars from Pa,Pb,Pc to BC, CA, AB concur at a point dP.
      As P moves on L, which is the envelope of the lines PdP ?

      APH
    • Angel
      Dear Antreas If the equation (barycentric) of L is px + qy + rz = 0, when P moves on the line L the lines PdP through the point L of first coordinate: L =(
      Message 2 of 3 , Feb 27, 2013
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        Dear Antreas


        If the equation (barycentric) of L is px + qy + rz = 0,
        when P moves on the line L the lines PdP
        through the point L' of first coordinate:

        L'=( a^6p(2p-q-r)+
        a^4(b^2(p(-5q+r)+2q(q+r))+c^2(p(q-5r)+2r(q+r)))-
        a^2(b^2-c^2)(b^2(2p^2+4q*r-p(5q+r))+c^2(-2p^2-4q*r+ p(q+5r)))+
        (b^2-c^2)^2(b^2(p-2q)-c^2(p-2r))(q-r) : ... : ... )

        L=Central line associated with a triangle center X then L'= orthojoin of X

        (The orthojoin of a point X other than X(6) is defined in ETC,
        notes just before X(1512))

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

        Why not on ETC the ORTHOJOIN OF X(12)?

        With (6-9-13)-search number -0.44020430340184767186343281...
        and first coordinate:

        (4a^8+a^6(-5b^2+2b*c-5c^2)
        -4a^5b*c(b+c)-a^4(b^2-c^2)^2
        +4a^3b*c(b-c)^2(b+c)+a^2(b-c)^4(b+c)^2
        +(b^2-c^2)^4)
        (a^6(b^2+c^2)-3a^4(b^4+c^4)
        -2a^3b^2c^2(b+c)+a^2(3b^6-b^4c^2+4b^3c^3-b^2c^4+3c^6)+
        2a*b^2(b-c)^2c^2(b+c)-(b^4-c^4)^2).


        Angel Montesdeoca

        --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
        >
        > Let ABC be a point, L a line and A',B',C' the orth. projections
        > of A,B,C in L, resp.
        > Let P be a point on L and Pa,Pb,Pc its reflections in AA',BB',CC', resp.
        > The perpendiculars from Pa,Pb,Pc to BC, CA, AB concur at a point dP.
        > As P moves on L, which is the envelope of the lines PdP ?
        >
        > APH
        >
      • Francois Rideau
        I thought it was the orthopole of L. So I was wrong Francois ... [Non-text portions of this message have been removed]
        Message 3 of 3 , Feb 28, 2013
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          I thought it was the orthopole of L.
          So I was wrong
          Francois

          On Thu, Feb 28, 2013 at 1:42 AM, Angel <amontes1949@...> wrote:

          > **
          >
          >
          > Dear Antreas
          >
          > If the equation (barycentric) of L is px + qy + rz = 0,
          > when P moves on the line L the lines PdP
          > through the point L' of first coordinate:
          >
          > L'=( a^6p(2p-q-r)+
          > a^4(b^2(p(-5q+r)+2q(q+r))+c^2(p(q-5r)+2r(q+r)))-
          > a^2(b^2-c^2)(b^2(2p^2+4q*r-p(5q+r))+c^2(-2p^2-4q*r+ p(q+5r)))+
          > (b^2-c^2)^2(b^2(p-2q)-c^2(p-2r))(q-r) : ... : ... )
          >
          > L=Central line associated with a triangle center X then L'= orthojoin of X
          >
          > (The orthojoin of a point X other than X(6) is defined in ETC,
          > notes just before X(1512))
          >
          > ----------------------------------
          >
          > Why not on ETC the ORTHOJOIN OF X(12)?
          >
          > With (6-9-13)-search number -0.44020430340184767186343281...
          > and first coordinate:
          >
          > (4a^8+a^6(-5b^2+2b*c-5c^2)
          > -4a^5b*c(b+c)-a^4(b^2-c^2)^2
          > +4a^3b*c(b-c)^2(b+c)+a^2(b-c)^4(b+c)^2
          > +(b^2-c^2)^4)
          > (a^6(b^2+c^2)-3a^4(b^4+c^4)
          > -2a^3b^2c^2(b+c)+a^2(3b^6-b^4c^2+4b^3c^3-b^2c^4+3c^6)+
          > 2a*b^2(b-c)^2c^2(b+c)-(b^4-c^4)^2).
          >
          > Angel Montesdeoca
          >
          >
          > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis wrote:
          > >
          > > Let ABC be a point, L a line and A',B',C' the orth. projections
          > > of A,B,C in L, resp.
          > > Let P be a point on L and Pa,Pb,Pc its reflections in AA',BB',CC', resp.
          > > The perpendiculars from Pa,Pb,Pc to BC, CA, AB concur at a point dP.
          > > As P moves on L, which is the envelope of the lines PdP ?
          > >
          > > APH
          > >
          >
          >
          >


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