We know that the inconic I(P) is the dual of the circumconic C(tP). The natural view of this
is that the duals of the points of C(tP) envelop the curve I(P). If X is a point on CI(tP), then
the contact point of the dual can be seen algebraically as the tX2-isoconjugate of P. But we
will see other ways to express it geometrically.
General duality is defined with respect to a conic C. The C-dual of a point is its polar in C,
the C-dual of a line is its pole in C. The usual dual is C-duality with respect to the diagonal
conic x2+y2+z2 = 0. The point-point mapping associated with duality can be expressed as
C(tP)-duality followed by the standard duality. This is defined at all points of the plane, and
has the correct effect on points of C(tP). The inverse mapping consists of I(P)-duality then
standard duality (as well as the obvious "standard duality followed by C(tP)-duality").
In algebraic language, this geometry leads to two useful mappings.
the map f(P,X) which takes X to the crosspoint of tX and P takes C(tP) to I(P),
the map g(P,X) which takes X to the tX-Ceva conjugate of tP takes I(P) to C(tP).
Theorem 1
(1) The points f(P,X),f(P,Y),f(P,Z) are collinear if and only if X,Y,Z are collinear.
(2) The points g(P,X),g(P,Y),g(P,Z) are collinear if and only if X,Y,Z are collinear.
Proof notes
Geometrically, this is more or less obvious; each duality preserves incidence.
Corollary
(1) f(P,X), f(P,Y) are antipodal if and only if X,Y,G are collinear.
(2) g(P,X), g(P,Y) are antipodal if and only if X,Y,G are collinear.
(3) f(P,X),f(P,Y),P are collinear if and only if X,Y,tP are collinear.
Notes on Corollary 1.1
(1) G is on C(tP) if and only if I(P) is a parabola (so contains its centre).
(2) G is on I(P) if and only if C(tP) is a parabola (so contains its centre).