Chasles's Theorem

If two projective Pencils of curves of orders $n$ and $n'$ have no common curve, the Locus of the intersections of corresponding curves of the two is a curve of order $n+n'$ through all the centers of either Pencil. Conversely, if a curve of order $n+n'$ contains all centers of a Pencil of order $n$ to the multiplicity demanded by Noether's Fundamental Theorem, then it is the Locus of the intersections of corresponding curves of this Pencil and one of order $n'$ projective therewith.

See also Noether's Fundamental Theorem, Pencil

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 33, 1959.