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Re: A configuration

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  • xpolakis
    Dear Jean-Pierre Thanks [APH] ... Variation: Lab := The Reflection of BB in AA Mab := The Parallel to Lab through B (instead of C). Lac := The Reflection of
    Message 1 of 5 , Jun 1, 2009
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      Dear Jean-Pierre

      Thanks

      [APH]
      > > Let ABC be a triangle and A'B'C' the orthic triangle.
      > >
      > > Denote:
      > >
      > > Lab := The Reflection of BB' in AA'
      > > Mab := The Parallel to Lab through C.
      > >
      > > Lac := The Reflection of CC' in AA'
      > > Mac := The Parallel to Lac through B.
      > >
      > > A* := Mab /\ Mac
      > >
      > > Similarly B*,C*.

      Variation:

      Lab := The Reflection of BB' in AA'
      Mab := The Parallel to Lab through B (instead of C).

      Lac := The Reflection of CC' in AA'
      Mac := The Parallel to Lac through C (instead of B).

      A* := Mab /\ Mac

      Similarly B*,C*.

      A*B*C* is the circumcevian triangle of H.

      [APH]
      > > 5. Let Na, Nb, Nc be the NPC centers of the triangles
      > > A*BC, B*CA, C*AB, resp.
      > > The triangles ABC, NaNbNc are perspective.
      > > Perspector?

      [JPE]
      > If U=X(54) is the isogonal conjugate of N and
      > V=X(140)=midpoint(ON), the perspector is P=-3U+4V
      > (this point lies on the line through the Lemoine point and
      > the two Corsican Imperial points)

      Corsican..........

      ÄéåôÞñåé áßìáôüò tïõ
      åéò ôáò öëÝâáò ôïõ ñáíßäá
      ï Êïñóéêáíüò ï Ý÷ùí
      ôïí Ôáàãåôïí Ðáôñßäá

      In the variation, we have: A'B'C'[=orthic], NaNbNc
      are perspective.

      Antreas
    • xpolakis
      [APH] ... [JPE] ... EQUIVALENTLY: Let ABC be a triangle, A1B1C1 the circumcevian triangle of H, and M1M2M3 the Medial triangle of ABC. Let A*,B*,C* be the
      Message 2 of 5 , Jun 3, 2009
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        [APH]
        > > Let ABC be a triangle and A'B'C' the orthic triangle.
        > >
        > > Denote:
        > >
        > > Lab := The Reflection of BB' in AA'
        > > Mab := The Parallel to Lab through C.
        > >
        > > Lac := The Reflection of CC' in AA'
        > > Mac := The Parallel to Lac through B.
        > >
        > > A* := Mab /\ Mac
        > >
        > > Similarly B*,C*.

        > > 5. Let Na, Nb, Nc be the NPC centers of the triangles
        > > A*BC, B*CA, C*AB, resp.
        > > The triangles ABC, NaNbNc are perspective.
        > > Perspector?

        [JPE]
        > If U=X(54) is the isogonal conjugate of N and
        > V=X(140)=midpoint(ON), the perspector is P=-3U+4V
        > (this point lies on the line through the Lemoine point and
        > the two Corsican Imperial points)

        EQUIVALENTLY:

        Let ABC be a triangle, A1B1C1 the circumcevian triangle
        of H, and M1M2M3 the Medial triangle of ABC.

        Let A*,B*,C* be the reflections
        of A1,B1,C1 in M1,M2,M3, resp. (symmetric points)

        Let Na, Nb, Nc be the NPC centers of the triangles
        A*BC, B*CA, C*AB, resp.
        The triangles ABC, NaNbNc are perspective.


        APH
      • xpolakis
        ... [JPE] ... From this we can make this generalization: Let ABC be a triangle, A1B1C1 the circumcevian triangle of point P, and M1M2M3 the Medial triangle of
        Message 3 of 5 , Jun 3, 2009
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          > [APH]
          > > > Let ABC be a triangle and A'B'C' the orthic triangle.
          > > >
          > > > Denote:
          > > >
          > > > Lab := The Reflection of BB' in AA'
          > > > Mab := The Parallel to Lab through C.
          > > >
          > > > Lac := The Reflection of CC' in AA'
          > > > Mac := The Parallel to Lac through B.
          > > >
          > > > A* := Mab /\ Mac
          > > >
          > > > Similarly B*,C*.
          >
          > > > 5. Let Na, Nb, Nc be the NPC centers of the triangles
          > > > A*BC, B*CA, C*AB, resp.
          > > > The triangles ABC, NaNbNc are perspective.
          > > > Perspector?
          >
          [JPE]
          > > If U=X(54) is the isogonal conjugate of N and
          > > V=X(140)=midpoint(ON), the perspector is P=-3U+4V
          > > (this point lies on the line through the Lemoine point and
          > > the two Corsican Imperial points)

          [APH]:
          >EQUIVALENTLY:
          >
          > Let ABC be a triangle, A1B1C1 the circumcevian triangle
          > of H, and M1M2M3 the Medial triangle of ABC.
          >
          > Let A*,B*,C* be the reflections
          > of A1,B1,C1 in M1,M2,M3, resp. (symmetric points)
          >
          > Let Na, Nb, Nc be the NPC centers of the triangles
          > A*BC, B*CA, C*AB, resp.
          > The triangles ABC, NaNbNc are perspective.


          From this we can make this generalization:

          Let ABC be a triangle, A1B1C1 the circumcevian triangle
          of point P, and M1M2M3 the Medial triangle of ABC.

          Let A*,B*,C* be the reflections
          of A1,B1,C1 in M1,M2,M3, resp. (symmetric points)

          Let Na, Nb, Nc be the NPC centers of the triangles
          A*BC, B*CA, C*AB, resp.

          Which is the locus of P such that:
          The triangles ABC, NaNbNc are perspective ?
          (O,I,H lie on the locus)

          Also:
          Which is the locus of P such that ABC, A*B*C*
          are perspective (or orthologic)?

          Which is the locus of P such that ABC, OaObOc
          are perspective?
          (Oa,Ob,Oc = Circumcenters of A*BC, B*CA, C*AB)

          Antreas
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