Coxeter-Figure 8.2 A (left)
8.21: If a quadrangle is inscribed in a conic, its diagonal triangle is self-polar.
Construct a conic, with 4 distinct points on the conic and connect the 4 points with six lines to form a complete quadrangle inscribed in a conic, as in Figure 8.2A (Left). The quadrangle is PQRS and the diagonal points are A=PS.QR, B=QS.RP, C=RS.PQ. The line AB meets the sides PQ and RS in points and such that are both conjugate to C. Thus the line AB, on which thy lie, is the polar of C. Similar construction would result in BC the polar of A, and CA the polar of B.