2452Anallagmatic cubics, part 8.
- Feb 2, 2001Anallagmatic 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
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
An anallagmatic cubic is a cubic with a pivot point which is always on the
line joining a point with its conjugate.
If L, M, N (from the inconic) are the coordinates of a point on the Steiner
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.
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.
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
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
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.
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.