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Euler Lines Reflected

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  • xpolakis
    Let ABC be a triangle and Ab,Ac the Orthogonal projections of A on the Reflections of BH,CH in AO, resp. Let La be the reflection of the Euler Line of AAbAc in
    Message 1 of 3 , May 8, 2009
      Let ABC be a triangle and Ab,Ac the Orthogonal projections
      of A on the Reflections of BH,CH in AO, resp.

      Let La be the reflection of the Euler Line of AAbAc
      in AbAc.

      Similarly Lb, Lc.

      Are the lines La,Lb,Lc concurrent?

      Antreas
    • xpolakis
      [APH] ... I think they are concurrent, and also the perpendiculars from A,B,C to La,Lb,Lc, resp. are concurrent. (on the circumcircle of ABC ?) Which is the
      Message 2 of 3 , May 8, 2009
        [APH]
        > Let ABC be a triangle and Ab,Ac the Orthogonal projections
        > of A on the Reflections of BH,CH in AO, resp.
        >
        > Let La be the reflection of the Euler Line of AAbAc
        > in AbAc.
        >
        > Similarly Lb, Lc.
        >
        > Are the lines La,Lb,Lc concurrent?

        I think they are concurrent, and also the perpendiculars
        from A,B,C to La,Lb,Lc, resp. are concurrent.
        (on the circumcircle of ABC ?)

        Which is the point of concurrence of La,Lb,Lc
        (1) wrt triangle ABC (2) wrt the orthic triangle of ABC
        (the lines AbAc, BcBa, CaCb bound the orthic triangle)

        Which is the point of concurrence of the perpendiculars?

        Antreas
      • xpolakis
        ... I have uploaded to Hyacinthos files the pdf file Francisco sent me with solutions: NAGEL POINT OF ORTHIC TRIANGLE X(185) and KIEPERT HYPERBOLA FOCUS X(110)
        Message 3 of 3 , May 8, 2009
          > [APH]
          > > Let ABC be a triangle and Ab,Ac the Orthogonal projections
          > > of A on the Reflections of BH,CH in AO, resp.
          > >
          > > Let La be the reflection of the Euler Line of AAbAc
          > > in AbAc.
          > >
          > > Similarly Lb, Lc.
          > >
          > > Are the lines La,Lb,Lc concurrent?
          >
          > I think they are concurrent, and also the perpendiculars
          > from A,B,C to La,Lb,Lc, resp. are concurrent.
          > (on the circumcircle of ABC ?)
          >
          > Which is the point of concurrence of La,Lb,Lc
          > (1) wrt triangle ABC (2) wrt the orthic triangle of ABC
          > (the lines AbAc, BcBa, CaCb bound the orthic triangle)
          >
          > Which is the point of concurrence of the perpendiculars?

          I have uploaded to Hyacinthos files the pdf file Francisco sent
          me with solutions: NAGEL POINT OF ORTHIC TRIANGLE X(185)
          and KIEPERT HYPERBOLA FOCUS X(110)

          I am wondering how could we get the GERGONNE point
          of the orthic triangle by a similar construction
          (ie by Euler lines reflected).

          Probably by a locus problem like this:

          Let ABC be a triangle, HaHbHc the orthic triangle,
          and P a point.

          Let Ab,Ac be the orthogonal projections of A
          on the Reflections of BH,CH in AP.

          Let La be the reflection of the Euler Line of AAbAc in HbHc.

          Similarly Lb,Lc.

          Which is the locus of P such that La,Lb,Lc are concurrent?

          APH
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