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2452Anallagmatic cubics, part 8.

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  • Steve Sigur
    Feb 2, 2001
      Anallagmatic cubics, part 8.

      ABC are the vertices of the reference triangle.
      w1, w2 are conjugate points (these notes default to isotomic conjugate with
      isogonic conjugates covered implicitely).

      A1,B1,C1 are the traces of w1 on the sides. Similarly for A2,B2,C2.
      G, Ga, Gb, Gc are the centroid and its harmonic associates (vertices of
      antimedial triangle).
      I, Ia, Ib, Ic are the incenter and its harmonic associates (the excenters).

      Q is the cubic with pivot w2 and Q' is the cubic with pivot w1.

      The complementary triangle is the medial triangle. The supplementary
      triangle is the cevian triangle of the incenter.

      The "complementaire' of a point (I will use the French word for this) is
      that point taken of the medial triangle. For example the complementaire of
      the circumcenter is the 9pt center.

      "Reciprocal" refers to the isotomic conjugate. "Inverse" is the isogonic

      The "reciprocal transversal" of a point is the dual line (l:m:n) -> lx+my+nz
      = 0

      An anallagmatic cubic is a cubic with a pivot point which is always on the
      line joining a point with its conjugate.


      Section 43

      If L, M, N (from the inconic) are the coordinates of a point on the Steiner
      ellipse then

      1/L + 1/M + 1/N = 0

      and the center of the conic is on the line at infinity.

      We see from this that:

      If the perspector (or center) describes the transverse reciprocal of the
      line harmonically associated with a point on the Steiner ellipse, the center
      (or perspector) of a circumconic traces a parabola circumscribed about the
      medial triangle whose diameters go through the reciprocal of the point on
      the Steiner ellipse.

      Section 44

      When this point is the Steiner point

      L(bb-cc) = M(cc-aa) = N(aa-bb)

      the center engenders the parabola

      sum xx(cc-aa)(aa-bb) + sum (bb-cc)^2 yz = 0

      and the perspector the line

      sum x/sin A.sin(B-C) = 0

      which is the inverse transversal of the Brocard axis.

      Section 45

      When the center moves on the Steiner ellipse, the reciprocal point (1/L :
      1/M : 1/N ) traces out the conic

      sum 1/(2x + y + z) = 0


      5 (xx+yy+zz) + 11(yz+zx+xy) = 0

      This conic is an ellipse concentric and homothetic to the Steiner ellipse,
      the ratio of homothety being 4.

      The point (L:M:N) traces the quartic

      5 sum yyzz + 11 xyz(x+y+z) = 0

      Section 46

      If the line that connects the perspector of a conic to its center passes
      through a fixes point (p:q:r) then these points engender a cubic

      | x y z |
      | |
      | x(x-y-z) y(y-z-x) z(z-x-y) | = 0
      | |
      | p q r |


      Q = sum p(y-z)/x = 0

      This cubic transforms by reciprocal points into the cubic

      Q = sum p xx (y-z) = 0

      We are able to see that this curve will be a cubic, as the points of a place
      which finds itself on a line through P are determined by this line and the
      conic to which it gives birth by the formulas of the transformation of
      section 33.

      Section 47

      And if we are given as a triangle of reference the medial triangle of ABC,
      we find that the cubic Q is, with respect to this triangle, an cubic
      anallagmatic by reciprocal points.

      Section 48

      This cubic has the same properties with respect to the medial triangle as
      does any anallagmatic cubic wrt its reference triangle.

      When a triangle is circumscirbed by a conic, the lines which join the
      perspector to the vertices and the center of gravity form form by the points
      of contact an anharmonic ratio equal to that of the lines which join the
      center of the conic to the vertices and the perspector.

      And since these properties are projective, we have also

      The point D is the pole of the line d with respect to conic c. Circumscribe
      a triangle to this conic and construct the trilinear pole of d with respect
      to the points of contack. P being the perspector of c with respect to the
      circumscribed triangle, the anharmonic ratio of the lines which join P to
      the points of cantack and to D is equal to that of the lines which join D to
      the vertices of the circumtriangle and to P.