a difficult proof!

The interchange theorem for projective conics - stronger version

If P and Q are distinct p-points in the interior of a p-conic C, then
there are precisely two projective transformations t which satisfy
(a) t(C) = C, and
(b) t interchanges P and Q.
and each such transformation has order 2

proof
Let the p-line PQ meet C in A,B, and let T be the pole of AB.

As in the proof of the two point theorem, we associate with P the
chord RS through P, the polar of the intersection of AB with the polar of P.
Likewise, we associate with Q, the chord R'S' through Q.

By La Hire's Theorem since RS and R'S' are the polars of p-points on AB,
they meet in T (the pole of AB).

Let RR' and SS' meet in W, and RS', R'S meet in V.
By the inscribed quadrilateral theorem , we have
(1) VW is the polar of T, and
(2) TV is the polar of W.

But AB is the polar of T, so, by (1), V,W lie on AB.
Also, by (2), V is the intersection of AB and the polar of W.

By the Corollary to the cross-ratio theorem,
one of V,W lies inside C, and one outside.

Now suppose that t satisfies (a) and (b).
As in the proof of the two point theorem,
and that of the interchange theorem, we see that
t interchanges A and B, and also {R,S} and {R',S'},
since it interchanges P and Q.

Since it interchanges A and B, it fixes AB, and hence fixes T, the pole of AB.
Since it maps {R,R',S,S'} to itself, it must also fix the set {V,W} - as the
other intersections of pairs of these p-points. But t maps C to itself, so
by the interior-exterior theorem, t must fix V and W since only one
of V,W is inside C.

We know that t maps R to one of R' and S', and S to the other.

Suppose that t maps R to R', and hence S to S'.
Also, t fixes W, so it maps RW to R'W.
It must therefore map R' to the other intersection of R'W and C, i.e. R.
Similarly, it must map S' to S.
Thus t2 fixes R,R',S,S'. Since these four p-points are on the p-conic C,
the three point theorem allows us to conclude that t2 is the identity.

The case where t(R) = S' is very similar (using V rather than W).


the interchange lemma