cross-ratios of the base points and base lines of two conics


Hi. Here's a problem of classical plane projective geometry
concerning certain invariant cross-ratios defined by two conics:

Let C and D be two plane conics (in sufficiently general position),
p1, p2, p3, p4 the four intersection points of C and D, and l1, l2,
l3, l4 the four common tangents (i.e., intersections of the dual
concics, C* and D*). Is the following statement true? After possibly
reordering, the cross-ratio of p1, p2, p3, p4 on C is equal to the
cross-ratio of l1, l2, l3, l4 on D* (which is the same as the
cross-ratio of the points of tangency of l1, l2, l3, l4 on D).

I found the following proof that the answer is "yes", but I'm not
fully convinced it's correct and, even if it, it's certainly *not* the
right way to do it:

Consider the set E of pairs (p,l) where p is on C and l is on D*
(i.e., l is a line tangent to D) such that p lies on l. Then E
(birational to) the intersection of two sufficiently general quadrics
in P^3, so E is a curve of genus 1. Now consider the two projections
f:E->C and g:E->D* (sending (p,l) to p and l respectively): both are
of degree 2. It is easy to see that the criticial values of f are p1,
p2, p3, p4, and those of g are l1, l2, l3, l4. So if we let gamma and
lambda be the respective cross-ratios, the j-invariant of E can be
computed as 256 (gamma^2-gamma+1)^3/(gamma^2(gamma-1)^2) (because the
critical points of f on E define a divisor of degree 4 which gives a
Weierstrass form of E) or exactly the same formula with lambda instead
of gamma: so both are equal up to action of S_4, QED.

(The curve E is classically introduced in the study of Poncelet's
theorem which deals, essentially, with order of the element of J(E)
given by the composition of the involutions defined by f and g. So
this doesn't come from nowhere. But it would seem very strange if
this couldn't be proved without appealing to the j-invariant of E.)

Bonus question: There are some other natural cross-ratios which we can
naturally associate to two general conics C and D, such as this:
parametrize the pencil to which C and D belong so that the three
singular members correspond to 0, 1 and infinity, then the parameters
of C and D in the pencil give natural cross-ratios associated to the
situation - or we can do the same for the duals. How can these
cross-ratios be related to those defined above?

--
David A. Madore
(david.madore@xxxxxx,
http://www.dma.ens.fr/~madore/ )
.