the parallel chords theorem

 The parallel chords theorem If F is a family of (at least two) parallel chords of C(Π), then there is a unique Π-diameter which bisects every member of F. Observations 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). Proof Let P' be the Π-centre of C(Π), and A'B' a member of F. Since F is a family of parallel segments on Π the common direction defines an ideal point Q' for Π. Note that, by the above observations, if C(Π) is a parabola, then we cannot have Q' = P', since there are no chords in the axial direction. Of course, if C(Π) is not a parabola, then P' is not ideal, so we cannot have Q' = P' in this case either. Let A,B,P,Q be the p-points corresponding to A',B',P',Q' respectively. Note that, as Q' is ideal for Π, Q lies on L(Π), the polar of P. Let M be the polar of Q with respect to C. By La Hire's theorem, P lies on M, so M embeds as a Π-diameter M' on Π. Let M cut AB at R. Since Q' ≠ P', M' is not the ideal line for Π. M' does contain the ideal point Q' so it contains no other ideal point. Thus, R' is a point on Π, and hence on A'B'. Again by La Hire's theorem, Q is on the polar of R. By the polar-chord theorem, (A,B,R,Q) = -1 as Q is on the polar of R. But Q' is ideal, so, by the mid-point theorem, R' is the mid-point of A'B'. Thus the Π-diameter M' bisects A'B'. Finally, as F contains at least two chords, there can be only one such Π-diameter.