The independence theorem
(1) If C(Π) is a parabola, then the Πcentre 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 Πcentre 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 halfturn 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 halfturn, A'B' is parallel to AB.
As C is not on AB, A'B' are distinct.
By the parallel chords theorem, there is a Πdiameter M' which
bisects AB and A'B'. But as AB, A'B' are distinct and parallel, there is
only one line through their midpoints.
Since r interchanges the midpoints of AB and A'B', r(M') = M'.
Thus, C lies on the Πdiameter M'.
Finally, since we can repeat this for any chord AB, C lies on all
Πdiameters. It follows that C is the Πcentre, the only point on
even two Πdiameters.

