Coxeter- Theorem 9.31 - Theorem 9.32 - Figure 9.3A

Pinyi Yao
Theorem 9.31: Desargues's Involution Theorem: Of the conics that can be drawn through the vertices of a given quadrangle, those which meet a given line (not through a vertex) do so in pairs of an involution. Proof: Let PQRS be the given quadrangle, and g the given line, meeting the sides PS, QS, QR, PR IN a,b,d,e, and any one of the conics in T and U. By regarding S, R, T ,U as four positions of a variable point on this conic, we see from 8.32 that the four lines joining them to P are projectively related to the four lines joining them to Q. Hence AETU are projective to BDTU. Since, by theorem 1.63, BDTU are projective to DBUT. Hence TU is a pair the involution (AD)(BE). Since this involution depends only on the quadrangle, all those conics of the pencil which intersect g (or touch g) determine pairs (or invariant points) of the same involution.