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Re: Incircle

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  • Antreas
    [APH] ... [Francisco] ... 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,
    Message 1 of 6 , Apr 10, 2013
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      [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
    • Francisco Javier
      You re right! P (L) = P (iL) (and P (iL) = P (L)) Francisco Javier.
      Message 2 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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