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Re: Incircle

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  • Francisco Javier
    You re right! P (L) = P (iL) (and P (iL) = P (L)) Francisco Javier.
    Message 1 of 6 , Apr 10, 2013
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      You're right! P'(L) = P"(iL) (and P'(iL) = P"(L))

      Francisco Javier.

      --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
      >
      > [APH]
      > > > Let ABC be a triangle, A'B'C' the pedal triangle of I
      > > > and A"B"C" the antipodal triangle of A'B'C' (ie the circumcevian
      > > > triangle of I wrt A'B'C'), and L a line.
      > > >
      > > > Denote:
      > > >
      > > > A1 = (Parallel to L through A') /\ (Pedal_Circle_of_I - A')
      > > > ie = the second intersection of the parallel to L
      > > > through A' and the Incircle.
      > > >
      > > > Similarly B1, C1.
      > > >
      > > > A2 = (Parallel to L through A") /\ (Pedal_Circle_of_I - A")
      > > > ie = the second intersection of the parallel to L
      > > > through A" and the incircle.
      > > >
      > > > Similarly B2, C2
      > > >
      > > > For which lines L the triangles:
      > > > 1. ABC, A1B1C1
      > > > 2. ABC, A2B2C2
      > > >
      > > > are perspective?
      > > >
      > > > If the answer is: For all lines L :
      > > >
      > > > Let L be a line passing through I. As L moves around
      > > > I, which are the loci of the perspectors?
      >
      > [Francisco]
      >
      > > In both cases the locus is the isogonal conjugate of the incircle, >that is a tricuspid quartic with the cusps at the vertices of the >triangle.
      >
      > I am wondering what perspectors we get by perpendicular
      > lines at I:
      >
      > Let L be line through I.
      >
      > Denote:
      > P'(L) = the perspector of ABC, A'B'C'
      > P"(L) = the perspector of ABC, A"B"C"
      >
      > iL = the perpendicular to L at I.
      >
      > Is this true?
      >
      > P'(L) = P"(iL) (and P'(iL) = P"(L))
      >
      > APH
      >
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