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[EMHL] Re: Perspective?

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  • 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 1 of 23 , Feb 13, 2013
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      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 2 of 23 , Feb 14, 2013
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        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 3 of 23 , Feb 14, 2013
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          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 4 of 23 , Feb 20, 2013
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            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 5 of 23 , Feb 20, 2013
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              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 6 of 23 , May 6, 2014
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                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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