The interchange theorem for projective conics
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.
proof
Let L be the p-line PQ. As P is inside C, L meets C twice - at A,B say.
Observe that L is determined by any two of the p-points P,Q,A,B.
Suppose that the projective transformation t maps C to itself and
maps Q to P, A to A or B. Then t also maps L to itself since it maps
AQ to AP or BP. It follows that t maps {A,B} to {A,B}.
Suppose that such a t maps A to A. Then it maps B to B.
By the cross-ratio theorem, (A,B,P,Q) = (A,B,t(P),P).
By Remarks(1),(3) (A,B,P,t(P)) = 1/(A,B,t(P),P).
Thus (A,B,P,Q) ≠ (A,B,P,t(P)), so, by Theorem PI1,
Q ≠ t(P).
Hence we cannot achieve a map of the required type this way.
Now suppose that t maps A to B. Then it maps B to A.
This time, we have (A,B,P,Q) = (B,A,t(P),Q) = (A,B,P,t(P)).
Then, by Theorem PI1, Q = t(P), as required.
By the two point theorem, there are precisely two such t.
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