The Π-diameter property
If A',B' (A'≠B') on C(Π) are such that A'B' is a Π-diameter of C(Π),
then the Π-centre of C(Π) is the mid-point of A'B'.
With the above notation, the Π-centre is ideal for Π if and only if the
plane conic C(Π) is a parabola. In such a case,
(1) the p-line L(Π) is the tangent to C at the Π-centre,
(2) each Π-diameter cuts C(Π) exactly once, and
(3) the Π-centre corresponds to the direction of the axis of C(Π).
Note first that, by the above observations, such a chord A'B' cannot
for a parabola.
Let P' denote the Π-centre of C(Π), and let A,B,P denote the p-points
with embeddings A',B',P' respectively. Then the p-line AB meets L(Π)
in a p-point Q, with embedding Q', say. As L(Π) corresponds to the
ideal line for Π, Q' is an ideal point for Π (so Q' ≠ P').
By the polar-chord theorem, (A,B,P,Q) = -1 as Q is on the polar of P.
But Q' is ideal, so, by the mid-point theorem, P' is the mid-point of A'B'.