- [APH]
> > Let ABC be a triangle, A'B'C' the pedal triangle of I

[Francisco]

> > 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?

> 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.

--- 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

>