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Radical Centers - perspective triangle

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  • Antreas Hatzipolakis
    Let ABC be a triangle and (Na),(Nb),(Nc) the NPCs of OBC, OCA,OAB, resp. Denote: Ra = the radical center of (O), (Nb), (Nc) Rb = the radical center of (O),
    Message 1 of 6 , Mar 24, 2013
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      Let ABC be a triangle and (Na),(Nb),(Nc) the NPCs of OBC, OCA,OAB, resp.

      Denote:

      Ra = the radical center of (O), (Nb), (Nc)

      Rb = the radical center of (O), (Nc), (Na)

      Rc = the radical center of (O), (Na), (Nb)

      The triangles ABC, RaRbRc are perspective.

      Perspector?

      APH
    • rhutson2
      Dear Antreas, The perspector is the isogonal conjugate of X(340) = X(92)-isoconjugate of X(323) = intersection of lines X(13)X*(14) and X(14)X*(13), where X* =
      Message 2 of 6 , Mar 24, 2013
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        Dear Antreas,

        The perspector is the isogonal conjugate of X(340) = X(92)-isoconjugate of X(323) = intersection of lines X(13)X*(14) and X(14)X*(13), where X* = inverse-in-circumcircle.

        Best regards,
        Randy

        --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
        >
        > Let ABC be a triangle and (Na),(Nb),(Nc) the NPCs of OBC, OCA,OAB, resp.
        >
        > Denote:
        >
        > Ra = the radical center of (O), (Nb), (Nc)
        >
        > Rb = the radical center of (O), (Nc), (Na)
        >
        > Rc = the radical center of (O), (Na), (Nb)
        >
        > The triangles ABC, RaRbRc are perspective.
        >
        > Perspector?
        >
        > APH
        >
      • Antreas Hatzipolakis
        Dear Randy The triangles OBC,OCA,OAB are special triangles (isosceles) and I think we can get perspectivities for other triads of circles respective to
        Message 3 of 6 , Mar 25, 2013
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          Dear Randy

          The triangles OBC,OCA,OAB are special triangles (isosceles)
          and I think we can get perspectivities for other triads of circles
          respective to BC,CA,AB.

          For example:

          Denote

          Ja = the excenter of the excircle respective to BC of the triangle OBC
          Jb = the excenter of the excircle respective to CA of the triangle OCA
          Jc = the excenter of the excircle respective to AB of the triangle OAB

          Ra = the radical center of (O), (Jb), (Jc)
          Rb = the radical center of (O), (Jc), (Ja)
          Rc = the radical center of (O), (Ja), (Jb)

          The triangles ABC, RaRbRc [I would bet that they] are perspective.

          Antreas


          On Mon, Mar 25, 2013 at 6:17 AM, rhutson2 <rhutson2@...> wrote:

          > **
          >
          >
          > Dear Antreas,
          >
          > The perspector is the isogonal conjugate of X(340) = X(92)-isoconjugate of
          > X(323) = intersection of lines X(13)X*(14) and X(14)X*(13), where X* =
          > inverse-in-circumcircle.
          >
          > Best regards,
          > Randy
          >
          >
          > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...>
          > wrote:
          > >
          > > Let ABC be a triangle and (Na),(Nb),(Nc) the NPCs of OBC, OCA,OAB, resp.
          > >
          > > Denote:
          > >
          > > Ra = the radical center of (O), (Nb), (Nc)
          > >
          > > Rb = the radical center of (O), (Nc), (Na)
          > >
          > > Rc = the radical center of (O), (Na), (Nb)
          > >
          > > The triangles ABC, RaRbRc are perspective.
          > >
          > > Perspector?
          > >
          > > APH
          > >
          >
          >
          >


          [Non-text portions of this message have been removed]
        • Antreas Hatzipolakis
          Here is a listing of this and other similar perspective (?) triangles. http://anthrakitis.blogspot.gr/2013/03/radical-centers-circumcircle-excircles.html aph
          Message 4 of 6 , Mar 25, 2013
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            Here is a listing of this and other similar perspective (?) triangles.

            http://anthrakitis.blogspot.gr/2013/03/radical-centers-circumcircle-excircles.html

            aph


            [APH]
            > Denote
            >
            > Ja = the excenter of the excircle respective to BC of the triangle OBC
            > Jb = the excenter of the excircle respective to CA of the triangle OCA
            > Jc = the excenter of the excircle respective to AB of the triangle OAB
            >
            > Ra = the radical center of (O), (Jb), (Jc)
            > Rb = the radical center of (O), (Jc), (Ja)
            > Rc = the radical center of (O), (Ja), (Jb)
            >
            > The triangles ABC, RaRbRc [I would bet that they] are perspective.
            >
            > Antreas
          • rhutson2
            Dear Antreas, They are indeed perspective (ETC search -17.765195072303639), though I could find no relationship to existing ETC centers, except that the line
            Message 5 of 6 , Mar 25, 2013
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              Dear Antreas,

              They are indeed perspective (ETC search -17.765195072303639), though I could find no relationship to existing ETC centers, except that the line though the perspector and the radical center of (Ja),(Jb),(Jc) (ETC search 5.603197179118705) passes through X(1790).

              Randy

              --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
              >
              > Dear Randy
              >
              > The triangles OBC,OCA,OAB are special triangles (isosceles)
              > and I think we can get perspectivities for other triads of circles
              > respective to BC,CA,AB.
              >
              > For example:
              >
              > Denote
              >
              > Ja = the excenter of the excircle respective to BC of the triangle OBC
              > Jb = the excenter of the excircle respective to CA of the triangle OCA
              > Jc = the excenter of the excircle respective to AB of the triangle OAB
              >
              > Ra = the radical center of (O), (Jb), (Jc)
              > Rb = the radical center of (O), (Jc), (Ja)
              > Rc = the radical center of (O), (Ja), (Jb)
              >
              > The triangles ABC, RaRbRc [I would bet that they] are perspective.
              >
              > Antreas
              >
              >
              > On Mon, Mar 25, 2013 at 6:17 AM, rhutson2 <rhutson2@...> wrote:
              >
              > > **
              > >
              > >
              > > Dear Antreas,
              > >
              > > The perspector is the isogonal conjugate of X(340) = X(92)-isoconjugate of
              > > X(323) = intersection of lines X(13)X*(14) and X(14)X*(13), where X* =
              > > inverse-in-circumcircle.
              > >
              > > Best regards,
              > > Randy
              > >
              > >
              > > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@>
              > > wrote:
              > > >
              > > > Let ABC be a triangle and (Na),(Nb),(Nc) the NPCs of OBC, OCA,OAB, resp.
              > > >
              > > > Denote:
              > > >
              > > > Ra = the radical center of (O), (Nb), (Nc)
              > > >
              > > > Rb = the radical center of (O), (Nc), (Na)
              > > >
              > > > Rc = the radical center of (O), (Na), (Nb)
              > > >
              > > > The triangles ABC, RaRbRc are perspective.
              > > >
              > > > Perspector?
              > > >
              > > > APH
              > > >
              > >
              > >
              > >
              >
              >
              > [Non-text portions of this message have been removed]
              >
            • rhutson2
              And if we substitute incircles for excircles, then RaRbRc is homothetic to ABC at (ETC search -0.387264139784864). Interestingly, the line through this
              Message 6 of 6 , Mar 25, 2013
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                And if we substitute incircles for excircles, then RaRbRc is homothetic to ABC at (ETC search -0.387264139784864). Interestingly, the line through this perspector and the radical center of (Ia),(Ib),(Ic) (ETC search 3.463390186710557) also passes through X(1790). Also, the two perspectors both lie on line X(25)X(41), and are {X(i),X(j)}-harmonic conjugates for (i,j) in {(25,1973), (41,2187), (42,205)}.

                Randy

                --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
                >
                > Dear Antreas,
                >
                > They are indeed perspective (ETC search -17.765195072303639), though I could find no relationship to existing ETC centers, except that the line though the perspector and the radical center of (Ja),(Jb),(Jc) (ETC search 5.603197179118705) passes through X(1790).
                >
                > Randy
                >
                > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@> wrote:
                > >
                > > Dear Randy
                > >
                > > The triangles OBC,OCA,OAB are special triangles (isosceles)
                > > and I think we can get perspectivities for other triads of circles
                > > respective to BC,CA,AB.
                > >
                > > For example:
                > >
                > > Denote
                > >
                > > Ja = the excenter of the excircle respective to BC of the triangle OBC
                > > Jb = the excenter of the excircle respective to CA of the triangle OCA
                > > Jc = the excenter of the excircle respective to AB of the triangle OAB
                > >
                > > Ra = the radical center of (O), (Jb), (Jc)
                > > Rb = the radical center of (O), (Jc), (Ja)
                > > Rc = the radical center of (O), (Ja), (Jb)
                > >
                > > The triangles ABC, RaRbRc [I would bet that they] are perspective.
                > >
                > > Antreas
                > >
                > >
                > > On Mon, Mar 25, 2013 at 6:17 AM, rhutson2 <rhutson2@> wrote:
                > >
                > > > **
                > > >
                > > >
                > > > Dear Antreas,
                > > >
                > > > The perspector is the isogonal conjugate of X(340) = X(92)-isoconjugate of
                > > > X(323) = intersection of lines X(13)X*(14) and X(14)X*(13), where X* =
                > > > inverse-in-circumcircle.
                > > >
                > > > Best regards,
                > > > Randy
                > > >
                > > >
                > > > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@>
                > > > wrote:
                > > > >
                > > > > Let ABC be a triangle and (Na),(Nb),(Nc) the NPCs of OBC, OCA,OAB, resp.
                > > > >
                > > > > Denote:
                > > > >
                > > > > Ra = the radical center of (O), (Nb), (Nc)
                > > > >
                > > > > Rb = the radical center of (O), (Nc), (Na)
                > > > >
                > > > > Rc = the radical center of (O), (Na), (Nb)
                > > > >
                > > > > The triangles ABC, RaRbRc are perspective.
                > > > >
                > > > > Perspector?
                > > > >
                > > > > APH
                > > > >
                > > >
                > > >
                > > >
                > >
                > >
                > > [Non-text portions of this message have been removed]
                > >
                >
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