We begin with the dual of the crossratio of four points on a line, namely the crossratio of
four lines through a point. We refer to such a configuration as a ppencil and the common
point as its vertex
lemma 1
If A,B,C,D are plines through a ppoint P and L a pline meeting
A,B,C,D in A,B,C,D respectively, then the crossratio (A,B,C,D)
is independent of the choice of L.
This allows us to make the
definition
If A,B,C,D are plines through a ppoint P, the crossratio (A,B,C,D)
is the crossratio of the ppoints where the plines meet any pline L
Since the crossratio of points is a projective invariant, we immediately have
lemma 2
The crossratio of a ppencil is a projective invariant
If we look at an embedding of a ppencil and pline L, we get a pencil on the embedding
plane with vertex P and a line L cutting the lines of the pencil at A,B,C,D. The crossratio
of the ppencil is equal to (A,B,C,D), and may be denoted by (PA,PB,PC,PD), though it
depends only on the lines, and not on the points chosen.
lemma 3
If a line cuts the lines of a pencil with vertex P at the points A,B,C,D,
then
(PA,PB,PC,PD) = sin(<APC)sin(<BPD)/sin(<BPC)sin(<APD).
lemma 4 Suppose that we have five points P,P',Q,R,S on a circle. If P,P' lie on the same side of RS, then as signed angles, then sin(<RPS) = sin(<RP'S). If P,Q lie on different sides of RS, then as signed angles, then sin(<RPS) = sin(<RQS).
proof
This can be interpreted as saying that as we move P on the


Chasles Theorem If A,B,C,D,P lie on a conic C, then the crossratio (PA,PB,PC,PD) is independent of the choice of P.
proof

Pascal's Theorem
If A,A',B'B',C,C' lie on a conic C and AB',A'B meet in P,
BC',B'C in Q and CA',C'A in R,
then P,Q,R are collinear.
proof
By Chasles' Theorem, (AA',AB',AC',AB) = (CA',CB',CC',CB).
As remarked above, these are equal to the crossratios of
the points which are intersections of the pencils with BA', BC'.
Thus (A',P,S,B) = (T,Q,C',B).
Now consider these crossratios as those of pencils through R.
(RA',RP,RA,RB) = (RC,RQ,RC',RB). As lines, RA'=RC, RA=RC',
so the pencils have three lines in common. Since their cross
ratios are equal, the lines RP,RQ are identical, as required.