  For two properly intersecting effective cycles in projective space X,Y, and their intersection product Z, the metric Bezout Theorem relates the degrees, heights of X,Y, and Z, as well as their distances and algebraic distances to a given point theta. Applications of this Theorem are in the area of Diophantine Approximation, giving estimates for approximation properties of Z with respect to $\theta$ against the ones of X, and Y. 
Diophantine Approximation on projective Varieties I: Algebraic distance   and metric B\'ezout Theorem