Coxeter- Theorem 9.41 - Theorem 9.42 - Figure 9.4A

Pinyi Yao
Combining 9.31 with 7.22, we see that the involution determined on g (Figure 9.3A) by the quadrangle PQRS is not only the Desargues involution determined by conics through P, Q, R, S but also the involution of conjugate points on g for the polarity (PQR)(Sg). Hence: Theorem 9.41: If two triangles have six distinct vertices, all lying on a conic, there is a polarity for which both triangles are self-polar. And conversely, Theorem 9.42: If two triangles, with no vertex of either on a side of the other, are self-polar for a given polarity, their six vertices lie on a conic and their six sides touch another conic.