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REGULAR POLYGON AND CONCURRENT CENTRAL LINES

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  • Antreas
    Let A1A2A3A4A5A6 be a regular hexagon and P a point. Name the triangles PA1A2, PA2A3, etc as 1,2,3,4,5,6. The Euler lines of the even triangles (ie 2,4,6) are
    Message 1 of 8 , Feb 20 1:44 PM
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      Let A1A2A3A4A5A6 be a regular hexagon and P a point.
      Name the triangles PA1A2, PA2A3, etc as 1,2,3,4,5,6.

      The Euler lines of the even triangles (ie 2,4,6)
      are concurrent at a point Q and of the odd triangles
      (ie 1,3,5) at a different point R.

      In General:

      Let A1A2A3....An, be a regular n-gon, with n =3k,
      and P a point. Name the triangles PA1A2, PA2A3, etc
      as 1,2,3,....n.

      Whose triangles the Euler lines are concurrent?

      APH
    • Antreas
      One more: Let ABC be an equilateral triangle, P a point, A;B C the cevian triangle of P, and A ,B ,C the reflections of A,B,C in A ,B ,C , resp. The Euler
      Message 2 of 8 , Feb 21 1:29 AM
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        One more:

        Let ABC be an equilateral triangle, P a point, A;B'C' the
        cevian triangle of P, and A",B",C" the reflections of
        A,B,C in A',B',C', resp.

        The Euler lines of AB"C", BC"A", CA"B"
        are concurrent.

        APH

        --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
        >
        > Another problem:
        >
        > Let ABC, A'B'C' be two arbitrary equilateral triangles.
        >
        > Denote:
        >
        > Ab, Ac = the reflections of A in BB', CC', resp.
        > Bc, Ba = the reflections of B in CC', AA', resp.
        > Ca, Cb = the reflections of C in AA', BB', resp.
        >
        > The Euler lines of AAbAc, BBcBa, CCaCb are concurrent.
        >
        > Corollary:
        >
        > Let ABC be an equilateral triangle and P a point.
        >
        > Denote:
        >
        > Ab, Ac = the reflections of A in BP, CP, resp.
        > Bc, Ba = the reflections of B in CP, AP, resp.
        > Ca, Cb = the reflections of C in AP, BP, resp.
        >
        > The Euler lines of AAbAc, BBcBa, CCaCb are concurrent.
        >
        > Proofs?
        >
        > APH
        >
        > --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
        > >
        > > Let ABC be an equilateral triangle and P a point.
        > >
        > > Which same central lines of the triangles PBC, PCA, PAB
        > > are concurrent for all P's ?
        > >
        > > The OH (Euler) lines? The OI lines? ......
        > >
        > > APH
        > >
        >
      • xpolakis
        ... One more: Let ABC be an equilateral triangle, P a point, A;B C the cevian triangle of P, and A ,B ,C the reflections of A,B,C in A ,B ,C , resp. The
        Message 3 of 8 , Apr 17, 2014
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          ---In Hyacinthos@yahoogroups.com, <anopolis72@...> wrote :

          One more:

          Let ABC be an equilateral triangle, P a point, A;B'C' the
          cevian triangle of P, and A",B",C" the reflections of
          A,B,C in A',B',C', resp.

          The Euler lines of AB"C", BC"A", CA"B"
          are concurrent.

          APH

          ****************************************************************************************

          In general, they are not.

          Which is the locus of P such that the Euler lines of AB"C", BC"A", CA"B" or
          the Euler lines of A"BC, B"CA, C"AB are concurrent?

          APH
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