If three conics pass through two given points and , then the lines joining the other two intersections of each
pair of conics
are concurrent at a point (Evelyn 1974, p. 15). The converse states that if two
conics
and
meet at four points ,
,
,
and ,
and if
and
are chords of
and ,
respectively, which meet on , 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 two of the points and are taken as the points
at infinity, then the theorem becomes that if two circles and pass through two points and on a conic , then the lines determined by the pair of intersections of
each circle with the conic are parallel (Evelyn 1974, p. 15).