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.