BarycentricCoordinates3

1. The tripole

This may be considered as a continuation of BarycentricCoordinates2.html .
There we discussed the trilinear polar of a point with respect to a triangle (we say triangle of reference ABC). There is an inverse construction of the Tripole point of a line with respect to a triangle.
Fixing the triangle of reference this map establishes a correspondence between the set of points and the set of lines represented in barycentric coordinates through the map:
Point P(x₀,y₀,z₀) (tripole) <------> line (1/x₀)x + (1/y₀)y + (1/z₀)z = 0 (tripolar).
Notation Often I write tr(P) and tr(L) both to denote the trilinear polar (line) of P or/and the tripole (point) of line L. The precise meaning follows from the context.

2. Conic generated by tripoles

The Tripols tr(L) of all lines ax+by+cz=0 through the point P(x₀,y₀,z₀) lie on a conic (c) with equation:
x₀/x + y₀/y + z₀/z = 0 <==> x₀*y*z + y₀*z*x + z₀*x*y = 0
passing through the vertices of the triangle of reference ABC.

Given the point P(x₀,y₀,z₀) consider all lines ax+by+cz=0 (with variable a,b,c) passing through that point i.e. satisfying
ax₀+by₀+cz₀=0 (1).
The tripoles of these lines are the points with respective coordinates (x',y',z') = (1/a,1/b,1/c), hence they satisfy
(x₀/x')+(y₀/y')+(z₀/z')=0 <==> x₀y'z' + y₀z'x' + z₀x'y' = 0 (2).
This is a quadratic equation in (x',y',z'), defining a conic passing through the vertices of the triangle (since the vertices are given by (1,0,0), (0,1,0), (0,0,1)) as claimed.

3. Isotomic conjugation in barycentrics

The transformation described in barycentrics by the formula:
t(x,y,z) = (1/x, 1/y, 1/z)
is defined for all points not lying on the side-lines of the triangle of reference ABC.
t is called the Isotomic Conjugation with respect to the triangle of reference ABC.

Denote by (P_A,P_B,P_C) the traces of point P(x,y,z) on the sides of ABC. From BarycentricCoordinates.html we know that P_AB/P_AC = -z/y. We deduce that the corresponding trace of point P'=t(P) on line BC divides BC in the ratio -y/z i.e. it is symmetric of P_A w.r to the midpoint of BC.
Two other properties of this transformation are:
[1] it is involutive i.e. t² = 1, and
[2] it maps lines ax+by+cz = 0 to conics through the vertices of ABC.
The first property is obvious and the last follows by setting x'=1/x, y'=1/y, z'=1/z. Then
ax+by+cz = 0 => a/x' + b/y' + c/z' = 0 i.e. ay'z' + bz'x' + cx'y' = 0.

4. The same conic described differently

The conic (c) x₀*y*z + y₀*z*x + z₀*x*y = 0, generated by the tripols of all lines through the point P(x₀,y₀,z₀), is also the isotomic image of the line x₀x + y₀y + z₀z =0.
This line is the trilinear polar tr(P') of the isotomic conjugate P'=t(P).

Nothing to prove. The claim follows from the definitions.

5. Additional remarks

[1] By the definition of A*,B*,C* follows (see Polar_Construction.html ) that the trilinear polar of P coincides with the polar of P with respect to conic (c). This is not true in general for t(P) and its trilinear polar ~P. Last line is not in general the polar of t(P) with respect to the conic c.
[2] Lines tr(P) and ~P=tr(t(P)) are related also by the isotomic transformation. This is discussed in IsogonalGeneralized.html .
[3] All conics circumscribing a triangle can be described/generated in these two ways. This double generation has many consequences studied in the references given below. Some insight in the relation of the two ways of generation of the conic (c) is contained in IsotomicConicOfLine.html .
[4] The notation used here conforms to the one used by Steve Sigur (see reference below):
tr(P), tr(L) denotes the tripolar (line), tripole (point),
t(P)=Q the isotomic conjugate of P,
~P = tr(t(P)) the dual line of P.

6. References