The Πdiameter property
If A',B' (A'≠B') on C(Π) are such that A'B' is a Πdiameter of C(Π),
then the Πcentre of C(Π) is the midpoint of A'B'.
Observations
With the above notation, the Πcentre is ideal for Π if and only if the
plane conic C(Π) is a parabola. In such a case,
(1) the pline L(Π) is the tangent to C at the Πcentre,
(2) each Πdiameter cuts C(Π) exactly once, and
(3) the Πcentre corresponds to the direction of the axis of C(Π).
Proof
Note first that, by the above observations, such a chord A'B' cannot arise
for a parabola.
Let P' denote the Πcentre of C(Π), and let A,B,P denote the ppoints
with embeddings A',B',P' respectively. Then the pline AB meets L(Π)
in a ppoint 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 polarchord theorem, (A,B,P,Q) = 1 as Q is on the polar of P.
But Q' is ideal, so, by the midpoint theorem, P' is the midpoint of A'B'.

