Now I think my "affine construction " of the dual line is false, so forget

it but all the rest is still true. PQR and its cofactor triangle beeing

perspective is the outcome of a general result about 2 triangles PQR and

P'Q'R' dual wrt some conic {Gamma}, here the equalizer. They are

perspective and if the perspector is O then the axis of perspective is the

polar of O wrt {Gamma}.

In case {Gamma} is a circle, it is clear that PQR and P'Q'R' are orthologic,

the 2 centers of orthology are the same point to say the center of the

circle {Gamma}, so fixed by the affine map f sending PQR to P'Q'R'.

It rest to look at the general case when {Gamma} is not a circle and to

understand why the center of {Gamma} is the fixed point of f and the fixed

lines of f are real and their directions conjugate wrt {Gamma}.

I notice that the perspector O of PQR and P'Q'R' is named "eigencenter" in

MathWorld, why?

Is there a connection with eigenvalues of some linear map?

Friendly

François

On 9/1/06, Francois Rideau <francois.rideau@...> wrote:

>

> Dear friends

> Given the affine reference triangle ABC and an arbitrary point D not on

> the sidelines of ABC, we look at the projective frame {A,B,C;D} of which D

> is the unit point.

> In this frame, every point M of the (extended projective) plane have

> homogeneous coordinates (x:y:z) and we have:

> A(1:0:0), B(0:1:0), C(0:0:1), D(1,1,1)

> In this frame, the dual line (L) of M(x:y:z) has equation:

> x.X + y.Y + z.Z = 0

> (L) is simply the polar line of M wrt the equalizer conic of equation: x.x+

> y.y + z.z = 0

> But as it is not very easy to construct a polar line wrt an imaginary

> conic,I give you (without proof!),a neat construction of (L) so you could

> check all what I said!

> Let {Gamma} the circumconic shaped with the equalizer, i.e sharing with

> the equalizer the same points at infinity.

> An equation of {Gamma} in the {A,B,C;D} frame is : y.z + z.x + x.y = 0

> Then {Gamma} is just the circumconic with center D, so very easy to

> construct.

> Let (L') be the polar line of M wrt {Gamma}, also easy to construct, then

> (L) is the image of (L') in the dilation of center D and ratio -1/2, also

> easy to construct.

> I am eager to have a more projective way to construct (L) from A, B, C, D,

> M.

> Please, help me!

> Of course with Cabri, I have a macro giving the fixed point and the

> invariant lines of any affine map f.

> For example, if you choose D as the circumcenter, then PQR and its

> cofactor triangle P'Q'R' are orthologic and perspective!

> Check it!

>

> Friendly

> François

>

>

>

> On 9/1/06, Francois Rideau <francois.rideau@...> wrote:

> >

> > Dear friends

> > We have already noticed the surprising fact that with the barycentric

> > definition of the cofactor triangle P'Q'R' of PQR wrt ABC that the fixed

> > point of the affine map f sending PQR to P'Q'R' is always the centroid G of

> > ABC.I also notice that the invariant lines of the map f are always real

> > (through G) and the directions of these 2 lines are conjugate wrt the

> > equalizer conic of barycentric equation: x.x + y.y + z.z =0 and so also

> > conjugate wrt every shaped conic with the equalizer, for instance the

> > Steiner ellipses.

> > Hence if ABC is equilateral, then triangles PQR and P'Q'R' are always

> > orthologic! Check that with your dynamic geometry software!

> > So in this special case, we could apply the Sondat theorem for triangles

> > PQR and P'Q'R' for they are also perspective.

> > By the way, I remember to have seen a recent and (russian?)

> > semi-synthetic proof of the Sondat theorem, different than the Sollertinski

> > projective synthetic complicated proof.

> > But I have lost it! Please, could you give me the reference!

> > I have also said the definition of cofactor triangle and most of their

> > properties are still valid in the plane with any homogeneous coordinates wrt

> > projective frame {A,B,C,D} where ABC is the affine reference triangle and D

> > an arbitrary unit point.

> > What can be said in this general case and look at the cases where D is

> > the incenter or the circumcenter.

> > Friendly

> > François

> >

> >

> > On 9/1/06, Francois Rideau < francois.rideau@...> wrote:

> > >

> > >

> > >

> > > Dear friends

> > > I just saw the definition of the unary cofactor triangle P'Q'R' of a

> > > triangle PQR wrt a triangle ABC in MathWorld.

> > > This definition uses trilinears and so the euclidian structure of the

> > > plane.

> > > What can be said if we replace trilinears by barycentrics, using only

> > > the affine structure of the plane?

> > > Is there a name for such triangle P'Q'R'? affine cofactor triangle?

> > > Some properties of the unary cofactor triangle are preserved. For

> > > example, PQR and P'Q'R' are still perspective. What's the name for the

> > > perpector O?

> > > The axis of perpective is the dual of O wrt ABC

> > > In a sense PQR and P'Q'R' are duals wrt ABC, for the dual of P wrt ABC

> > > is Q'R', the dual of P' wrt ABC is QR and so one...

> > > Besides, the centroid G of ABC is the fixed point of the affine map

> > > sending P to P', Q to Q', R to R' and so one...

> > > I think all these properties are already known and I need some

> > > references.

> > > Thanks in advance

> > > Friendly

> > > PS

> > > Of course, all these definitions and properties are still valid in a

> > > projective plane, using homogeneous coordinates wrt some projective frame

> > > {A,B,C,D} where D is an arbitrary unit point.

> > > François

> > >

> >

> >

>

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