Anallagmatic cubics, part two.
This continues my notes from Cubiques Anallagmatiques by Ad. Mineur.
I really like this little book. As I work through it I am posting my notes.
There were three ways of approaching cubics that I knew before. Peter Yff defined cubics as curves where the conjugates lie on a straight line through a common point. John Conway defines them as special loci: points for which, say, the pedal triangle of a point is the cevian triangle of another point. A third way is to start with the group table, using algebra developed in the last century.
Mineur's presentation is different, emphasizing the relation between a pair of points, a pair of lines, a pair of cubics, and a pair of conics. I like this interplay between the four types of structures, particularly as I have been feeling that the interplay between cubics and conics was important. The work of Bernard and Jean-Pierre in our e-group made me think this more strongly.
In addition Miniur emphasizes that the above pairs of geometric objects occur twice, once in a form related to the centroid, for which barycentrics and are natural and the inverse relation is the isotomic conjugate; and once in a form related to the centroid for which trilinears are natural and the inverse relation is the isogonic conjugate. Hence two pairs of structures of pairs of objects. In The Triangle Book the relation that will take the isotomic conjugate to the isogonic conjugate is the "pro" operation that John and I think is so important.
The notes continue....
Section 5
The lines lx + my + nz = 0 and l'x + m'y + n'z = 0 are paired with cubics
Q: ... + m y (zz-xx) + ... = 0
and
Q': ... + m' y (zz-xx) + ... = 0
These two cubics meet in 9 points, 7 of which are ABC G and the three harmonic associates of G.
The other two points are given as solutions of
x(yy-zz) y(zz-xx)
--------- = -------- = ...
mn' - n'm n'l - l'n
The points ABC also satisfy these equations.
These two point lie on the line x(mn'-nm') + y(nl'-n'l) + z(lm'-ml') = 0, which passes through (l:m:n) and (l':m':n') and which is harmonically associated with [this statement justifies the rather complicated looking formula a couple of line above.]
( : 1/(nl'-n'l) : )
which is the 4th point of intersection of the circumconics
l/x + m/y + n/z = 0 and l'/x + m'/y + n'/z = 0
Section 6
if ll' = mm' = nn' then the lines D1 and D2 are reciprocal and the points
w1 = (1/l:1/m:1/n) and w2 = (l:m:n)
are the same as
w1' = (l': m': n') and w2' = (1/l':1/m':1/n')
These points are on the cubics Q and Q'. Q and Q' will always mean this particular case so that every object and its primed version are reciprocal.
Section 7
The tangents to the cubic Q at the vertices ABC are
lx = my = nz which concur at w1.
Similarly the tangents to the cubic Q' at the vertices ABC are
x/l = y/m = z/n which concur at w2.
This gives the theorem
The tangents at ABC to two reciprocal cubics are isotomic lines, [and meet on both cubics].
Section 8
Substituting succesively x = 0, y = 0, then z = 0 into the cubic Q shows that the traces of w1 on the edges are on the cubic. The traces of w2 are similarly on Q', the corresponding traces are isotomic points.
Section 9
There are many harmonic relations among the lines. The equalities
B (A2 A G Gc) = A (A2 w1 G Gc) = -1
C (A2 A G Gb) = A (A2 w1 G Gb) = -1
show that the conics G A B Gc A2 and G A C Gb A2 are tangent at A to Q and have AW1 as the common tangent. Here A2 is the trace of w2 on BC and Gc is the C harmonic associate of G. [The notation B (A2 A G Gc) is the "cross ratio" of lines BA2 BA BG BGc , the fact that this equals -1 shows that they are an harmonic range].
We then have a theorem
A line from A meets BC in Y. On this line take X, Xa to be harmonic conjugates. Then all lines drawn through Y meet the sides AC, AB in two points B', C' of a conic circumscribed about the quadrilateral B X C Xa. The points of intersection of X B' with Xa C' and that of X C' with Xa B' form the line harmonically conjugate to AY in the angle BAC.
Section 10
These results establish that the reciprocal points w1 and w2 are associated with
The reciprocal transversal lines D and D'
The reciprocal conics C and C'
and the reciprocal cubics Q and Q'
Section 11
If (x,y,z) is on the cubic then
(x,y,z) and (1/x,1/y,1/z) are colinear with (l,m,n).
Theorem:
If the line with joins reciprocal points always passes through a fixed point, the reciprocal points trace out a cubic.
We call the fixed point the pole of the cubic. [Note: another common term is "the pivot"].
Section 12
The line which joins the reciprocal points turns, by reciprocal transformation, into a conic that goes through ABC and the two points. This has the consequence that all lines that go through w2 (the pole of Q) determine on the cubic Q, two reciprocal points situated on a conic circumscribed to ABC and passing through w1. [I call this the Mineur conic]
Hence one can consider the cubic Q as the locus of the points of intersection of a line which turns around a fixed point and the circumconic formed from the reciprocal of the line.
This gives a method for constructing the cubic as a locus.
Section 13
Let the mobile line take the postion w1w2, remembering that the cubic goes through these points and that they are reciprocal.
w1w2 is tangent to the cubic Q at w2 and to Q' at w1.
Section 14
Transform this by reciprocal points and the line w1 w2 becomes a conic ABC w1 w2, which is tangent to Q at w1 and Q' at w2.
Section 15
Now let the mobile line (pivoting around w2, the pole of Q) take one of the positions w2 G, w2 Ga, w2 Gb, w2 Gc, where Ga is the "a" harmonic assiciate of G (the vertex of its preCevian triangle). Note that each of these points is its own reciprocal, that it is a double point and the line is tangent to the cubic at that point. We now form conics like ABCG w1, which are also tangent to the cubic Q at these points, the lines w2 G being tangents.
Section 16
When the mobile line coincides with w2 A, we conclude that w1 A is tangent to the cubic and that the cubic goes through the trace of w2 on BC. Same for the other vertices.
Section 17
A summary of results with nothing new.
... more here ...