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Re: [EMHL] Re: Perspective?

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  • Antreas Hatzipolakis
    As locus problem: Let ABC be a triangle, P a point and PaPbPc the pedal triangle of P. Denote: (N1), (N2), (N3) = the reflections of (N) in PaO, PbO, PcO,
    Message 1 of 23 , Feb 12, 2013
      As locus problem:

      Let ABC be a triangle, P a point and PaPbPc the pedal triangle of P.

      Denote:
      (N1), (N2), (N3) = the reflections of (N) in PaO, PbO, PcO, resp.

      A'B'C' = the triangle bounded by the radical axes of
      ((O),(N1)), ((O),(N2)),((O),(N3)).

      Which is the locus of P such that A'B'C', N1N2N3 are perspective?

      Euler Line (?) + ??

      APH

      GENERALIZATION:

      >
      > Denote:
      >
      > PaPbPc = the pedal triangle of a point P on the Euler line
      >
      >
      > (N1), (N2), (N3) = the reflections of (N) in PaO, PbO, PcO, resp.
      >
      > A'B'C' = the triangle bounded by the radical axes of
      > ((O),(N1)), ((O),(N2)),((O),(N3)).
      >
      > A'B'C', N1N2N3 are perspective.
      >
      > True???
      >
      >
      >


      [Non-text portions of this message have been removed]
    • Antreas
      GENERALIZATION Let ABC be a triangle, Q a point, QaQbQc the pedal triangle of Q, (X) the circumcircle of QaQbQc (=pedal circle of Q), P a point on the OQ line
      Message 2 of 23 , Feb 13, 2013
        GENERALIZATION

        Let ABC be a triangle, Q a point, QaQbQc the pedal triangle of Q,
        (X) the circumcircle of QaQbQc (=pedal circle of Q), P a point on
        the OQ line and PaPbPc the pedal triangle of P.

        Denote:

        (X1), (X2), (X3) the reflections of (X) in OPa,OPb,OPc, resp.

        A'B'C' = the triangle bounded by the radical axes of ((O),(X1)),((O),(X2)),((O),(X3))

        Conjecture:
        The triangles A'B'C', X1X2X3 are perspective.

        Figure:
        http://anthrakitis.blogspot.gr/2013/02/reflecting-pedal-circle.html

        Antreas

        [APH]
        >
        > Denote:
        >
        > PaPbPc = the pedal triangle of a point P on the Euler line
        >
        > (N1), (N2), (N3) = the reflections of (N) in PaO, PbO, PcO, resp.
        >
        > A'B'C' = the triangle bounded by the radical axes of
        > ((O),(N1)), ((O),(N2)),((O),(N3)).
        >
        > A'B'C', N1N2N3 are perspective.
        >
        > True???
      • Antreas
        Let HaHbHc be the orthic triangle of ABC, (N1),(N2),(N3) the reflections of the NPC (N) in the sidelines HbHc, HcHa, HaHb of HaHbHc resp. and and A B C the
        Message 3 of 23 , Feb 14, 2013
          Let HaHbHc be the orthic triangle of ABC, (N1),(N2),(N3) the
          reflections of the NPC (N) in the sidelines HbHc, HcHa, HaHb of
          HaHbHc resp. and and A'B'C' the triangle bounded by the radical
          axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.

          Are the triangles ABC, A'B'C' perspective ?

          aph

          [APH]
          > 2. Let HaHbHc be the orthic triangle, (N1),(N2),(N3) the reflections
          > of the NPC (N) in the altitudes HHa,HHb,HHc,
          > resp. and and A'B'C' the triangle bounded by the radical axes of
          > ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
          >
          > Are the triangles HaHbHc, A'B'C' perspective ?
        • Angel
          Dear Antreas, The triangles ABC, A B C are perspective. Perspector: (Conway notations) (SA^2-3S^2)(S^2-SB^2)(S^2-SC^2))/SA: ... : ... Best regards. Angel M.
          Message 4 of 23 , Feb 14, 2013
            Dear Antreas,

            The triangles ABC, A'B'C' are perspective.

            Perspector: (Conway notations)

            (SA^2-3S^2)(S^2-SB^2)(S^2-SC^2))/SA: ... : ...

            Best regards.
            Angel M.


            --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
            >
            > Let HaHbHc be the orthic triangle of ABC, (N1),(N2),(N3) the
            > reflections of the NPC (N) in the sidelines HbHc, HcHa, HaHb of
            > HaHbHc resp. and and A'B'C' the triangle bounded by the radical
            > axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
            >
            > Are the triangles ABC, A'B'C' perspective ?
            >
            > aph
          • Antreas
            I am not sure If I have already posted this: Let ABC be a triangle, (N1),(N2),(N3) the reflections of (N)[=NPC] in OA,OB,OC, resp. and A B C the triangle
            Message 5 of 23 , Feb 20, 2013
              I am not sure If I have already posted this:

              Let ABC be a triangle, (N1),(N2),(N3) the reflections of
              (N)[=NPC] in OA,OB,OC, resp. and A'B'C' the triangle bounded
              by the radical axes of ((O),(N1)),((O),(N2)),((O),(N3)).

              Are N1N2N3, A'B'C' perspective?

              APH
            • Antreas Hatzipolakis
              One more.... HaHbHc = the orthic triangle. (N1) = the circle (Ha, HaN) ie the circle with center Ha and radius HaN =R/2 Similarly (N2),(N3) A B C = the
              Message 6 of 23 , Feb 20, 2013
                One more....

                HaHbHc = the orthic triangle.
                (N1) = the circle (Ha, HaN) ie the circle with center Ha and radius HaN =R/2
                Similarly (N2),(N3)

                A'B'C' = the triangle bounded by the radical axes of
                ((O),(N1)),((O),(N2)),((O),(N3)).

                Are the triangles HaHbHc, N1N2N3 perspective?

                aph


                On Wed, Feb 20, 2013 at 1:29 PM, Antreas <anopolis72@...> wrote:

                > **
                >
                >
                > I am not sure If I have already posted this:
                >
                > Let ABC be a triangle, (N1),(N2),(N3) the reflections of
                > (N)[=NPC] in OA,OB,OC, resp. and A'B'C' the triangle bounded
                > by the radical axes of ((O),(N1)),((O),(N2)),((O),(N3)).
                >
                >
                > Are N1N2N3, A'B'C' perspective?
                >
                > APH
                >
                > __
                >


                [Non-text portions of this message have been removed]
              • Antreas Hatzipolakis
                Let ABC be a triangle with excenters Ia,Ib,Ic. The NPC of AHIa interscts the excircle (Ia) at A other than the Feuerbach point Fa. The NPC of BHIb
                Message 7 of 23 , May 6, 2014

                  Let ABC be a triangle with excenters Ia,Ib,Ic.

                  The NPC of  AHIa interscts the excircle (Ia) at A' other than the
                  Feuerbach point Fa.

                  The  NPC of  BHIb intersects the excircle (Ib) at B' other than the
                  Feuerbach point Fb,

                  The  NPC of  CHIc intersects the excircle (Ic) at C' other than the
                  Feuerbach point Fc.

                  Are the triangles ABC, A'B'C' perspective?

                  In any case, has the triangle A'B'C' any interesting properties?

                  APH

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