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17837Re: A configuration

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  • xpolakis
    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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