Given a line $p$ and a line segment $\overline{MN}$ on $p$, and two circles $k_1$ and $k_2$, construct a line segment $\overline{AB}$ with endpoints on the circles so that $\overline{AB}$ is parallel to $p$ and the same length as $\overline{MN}$.
I can see that there is no solution if the circles are on different sides of the line $p$. But what would be the approach to do the construction when a solution exists?
Just translate $k_1$ using $NM$ or $MN$ as vector in the direction of $k_2$
where this translated circle cuts $k_2$ are the points where the endpoints of $AB$ are (there can be two of them)