Browse Groups

• [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
View Source
[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
• You re right! P (L) = P (iL) (and P (iL) = P (L)) Francisco Javier.
Message 1 of 6 , Apr 10
View Source
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
>
Your message has been successfully submitted and would be delivered to recipients shortly.
• Changes have not been saved
Press OK to abandon changes or Cancel to continue editing
• Your browser is not supported
Kindly note that Groups does not support 7.0 or earlier versions of Internet Explorer. We recommend upgrading to the latest Internet Explorer, Google Chrome, or Firefox. If you are using IE 9 or later, make sure you turn off Compatibility View.