\[h(X\cap Y) = \deg(X)h(Y) + \deg(Y)h(X) + O(1)\]
where $X$ and $Y$ are arithmetic cycles varying in certain projective families, and $O(1)$ is a function bounded independently of the choice of $X$ and $Y$, and $h$ is a logarithmic height function. I will also present an application of this formula to an analogue of Hilbert's Irreducibility Theorem for intersections of curves in ${\mathbf P}^2$.