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17961[EMHL] CONJECTURE (Re: More Reflections in bisectors)

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  • xpolakis
    Jul 4, 2009
      [APH]
      > > > Let La, Lb, Lc be the reflections of PA', PB', PC'
      > > > in the bisectors AI, BI, CI, resp.
      > > >
      > > > Which is the locus of P such that the triangles
      > > > ABC, Triangle bounded by (La, Lb, Lc) are
      > > > parallelogic ?
      > > >
      > >
      > > How about if A'B'C' is the orthic triangle?
      > >
      > > Which is the locus of P?

      [ND]
      > The locus is the linf +
      > the conic with center X(1112)
      > that passes through the vertices of the cevian
      > triangles of X(4) Orthocenter and
      > X(648) the trillinear pole of Euler line.
      > See the notes in ETC for X(1112).

      Dear Nikos

      It is nice! Thanks.

      Probably there are also nice results for pedal triangles
      of other points.

      For the pedal triangle of I, for example.

      That is:

      Let ABC be a triangle, and A'B'C' the pedal triangle of I.

      Let La, Lb, Lc be the reflections of PA', PB', PC'
      in AI, BI, CI, resp.

      Which is the locus of P such that the triangles
      ABC, Triangle bounded by (La, Lb, Lc) are
      parallelogic ?

      Antreas
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