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Re: Cofactor Triangle

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  • Francois Rideau
    Dear friends 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
    Message 1 of 4 , Sep 1, 2006
      Dear friends
      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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