If two intersections of each pair of three conics S_1, S_2, and S_3 lie on a conic S_0, then the lines joining the other two intersections of each pair are concurrent (Evelyn et al. 1974, pp. 23 and 25). The dual theorem states that if two common tangents of each pair of three conics touch a fourth conic, then the remaining common tangents of each pair intersect in three collinear points .

conic section | three conics theorem