If three conics pass through two given points Q and Q', then the lines joining the other two intersections of each pair of conics P_(i j) P_(i j)^, are concurrent at a point X. The converse states that if two conics E_2 and E_3 meet at four points Q, Q', P_1, and Q_1, and if P_2 Q_2 and P_3 Q_3 are chords of E_3 and E_2, respectively, which meet on P_1 Q_1, then the six points lie on a conic. The dual of the theorem states that if three conics share two common tangents, then their remaining pairs of common tangents intersect at three collinear points. If the points Q and Q' are taken as the points at infinity, then the theorem reduces to the theorem that radical lines of three circles are concurrent in a point known as the radical center.

conic section | four conics theorem | radical center