the independence theorem

The independence theorem
(1) If C(Π) is a parabola, then the Π-centre is the ideal point for Π
corresponding to the axial direction for the plane conic.
(2) If C(Π) is an ellipse or hyperbola, then the Π-centre is the affine
centre of the plane conic.

Proof of part(2)
As the plane conic C' = C(Π) is an ellipse or hyperbola, its affine
symmetry group contains a half-turn r. Let C be the centre of r.

Let AB be a chord of C' which does not pass through C, and
let A' = r(A), B' = r(B).
As r maps C' to itself, A'B' is also a chord.
As r is a half-turn, A'B' is parallel to AB.
As C is not on AB, A'B' are distinct.

By the parallel chords theorem, there is a Π-diameter M' which
bisects AB and A'B'. But as AB, A'B' are distinct and parallel, there
is only one line through their mid-points.
Since r interchanges the mid-points of AB and A'B', r(M') = M'.
Thus, C lies on the Π-diameter M'.

Finally, since we can repeat this for any chord AB, C lies on all
Π-diameters. It follows that C is the Π-centre, the only point on
even two Π-diameters.

central conics revisited