Chasles's theorem states that if two projective pencils of curves of orders 
and 
have no common curve, the locus of the intersections of
corresponding curves of the two is a curve of order 
through all the centers of either pencil.
Conversely, if a curve of order 
contains all centers of a pencil
of order 
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
projective therewith.
Chasles's Theorem
See also
Noether's Fundamental Theorem, PencilExplore with Wolfram|Alpha
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 33, 1959.Referenced on Wolfram|Alpha
Chasles's TheoremCite this as:
Weisstein, Eric W. "Chasles's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChaslessTheorem.html