Coxeter-Theorem 8.32

8.32. STEINER'S THEOREM: Let line x and y join a variable point on a conic to two fixed points on the same conic; then x and y is a projectivity. Construct tangents p and q, at the fixed points P and Q, meet in D, the pole of PQ as in Figure 8.3 A. Let c be a fixed line through D (but not through P or Q), meeting x in B, and y in A. By 8.31 (SEYDEWITZ'S THEOREM), BA is a pair of the involution of conjugate points on c. Hence, when the point R=x.y varies on the conic, we have xprojective to B, B projective to A , and A projective y, as desired.