Study's Theorem

Given three curves $\phi_1$, $\phi_2$, $\phi_3$ with the common group of ordinary points $G$ (which may be empty), let their remaining groups of intersections $g_{23}$, $g_{31}$, and $g_{12}$ also be ordinary points. If $\phi'_1$ is any other curve through $g_{23}$, then there exist two other curves $\phi'_2$, $\phi'_3$ such that the three combined curves $\phi_i\phi'_i$ are of the same order and Linearly Dependent, each curve $\phi'_k$ contains the corresponding group $g_{ij}$, and every intersection of $\phi_i$ or $\phi'_i$ with $\phi_j$ or $\phi'_j$ lies on $\phi_k$ or $\phi'_k$.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 34, 1959.