Mineur conics
I came across a way of geometric thinking in Mineur's book, found here. This book contains a conic of particularly high symmetry, generated as the conjugate of the line joining a point with its conjugate. It has become a central feature in my understanding of triangle conics, and indeed the plane of the triangle. As the idea seems to be little known, I would like to tell people about it.
The trick is to maintain symmetry between a point and its conjugate, which Mineur calls "anallagmatic symmetry." I alternate between considering this a true symmetry or simply a more aestheic way of doing geometry. We will use the isotomic conjugate until the end when other conjugations will be discussed.
The Mineur conic is important in the theory of pivotal cubics. This will be discussed at the end.
Notation:
The operator t represents the isotomic conjugate, m and d are the medial and antimedial opeations.
For a circumconic the center is found from the perspector using all three operations: center = mtd perspector.
The dual of a point P is written as ~P. The tripolar of P is the dual of the isotomic conjugate.
Anallagmatic properties of points and lines
If a point and its conjugate be interchanged and an object derived from them be unchanged, we say that the object has anallagmatic symmetry. I name these objects after Mineur.
Mineur point = ~P•~tP
Mineur line = ~P — ~tP
The Mineur conic is introduced below.
Let P be a point in the plane of the triangle and P' = tP, its conjugate. From these two form mP and mP'. The first very interesting property is that
The isotomics of these four points are all on a line.
P, tmP', tmP, P' are on the line between P and P', between a point and its conjugate, proved here. This line is anallagmatically symmetric. I call it the Mineur line. Letting P = ( l : m : n ) its equation is
l (m2 – n2) x + m ( n2 – l2) y + n ( l2 – m2) z = 0
The isotomic of this line is a circumconic, the Mineur conic, whose perspector is the dual of the line.
l (m2 – n2) / x + m ( n2 – l2) / y + n ( l2 – m2) / z = 0
In addition to A, B, C, the Mineur conic contains the isotomics of the 4 colinear points are on this conic: P, mP, P', mP'. This single fact is very impressive. It guarantees that the conic will have interesting points, at least if P is interesting.
The tripolars of P, mP, mP', P' concur at the Mineur point which is the perspector of the Mineur conic and the dual of the Mineur line.
The equations of two of these lines are
of P': l x + m y + n z = 0
of P: x/l + y/m + z/n = 0.
which concur at the Mineur point
( l (m2 – n2) / x : m ( n2 – l2) : n ( l2 – m2) )
the dual of the Mineur line. Note that this expression is unchanged under the substitution (l:m:n) –> (1/l:1/m:1/n) which is the statement of anallagmatic symmetry for isotomic conjugacy.
Here is a picture showing the Mineur point, line and conic.
The parallelogram in this figure is the affine parallelogram for which the Mineur line is a diagonal. The center of the Mineur conic lies on the other diagonal.
Examples:
If P = No, the Nagel point, the Mineur conic is the Feuerbach hyperbola. If P = H, it is the Jerabek hyperbola.
This one is particularly intersesting since it is a double Mineur conic since the conjugates D, H reside on it as do the isogonal conjugates O and H.
Polar lines and Polar Conics of a Pivotal Cubic
This section is more difficult. I include it just to show the connection of the Mineur conic to certain cubics.
If P0 is a point in the plane of a pivotal cubic K = 0, then there are two polar equations
P0 · DK(P) = 0 and P · DK(P0) = 0
where D indicates the derivative (in the multivariable sense; ie, the gradient). The first is the polar conic and the second is the polar line. If P0 is on the cubic then the line and the conic are tangent to the cubic at P0.
For a pivotal isotomic cubic the polar line is
x ( 2x0(my0 – nz0) + l (y02 – z02) ) + cyclic = 0
The polar conic is
x0[ 2x( my – nz) + l(y2 –z2) ] + cyclic = 0
This gist of all this is that
If P is the pivot of a cubic, then the tangent conic at P' = tP is the Mineur conic.