I would like to know how to solve this problem:
If you have overlapping circles $A$ and $B$ (with $B$'s area is at least half the area of $A$), what are the points of intersection of circle $A$ when the area of intersection between $A$ and $B$ is equal to half the area of $A$ (or any portion of $A$)?
The area of the intersection between your circles is given by: $$ A=r^2\cos^{-1}{d^2+r^2-R^2\over2dr} +R^2\cos^{-1}{d^2+R^2-r^2\over2dR} -{1\over2}\sqrt{(-d+r+R)(d+r-R)(d-r+R)(d+r+R)}, $$ where $R$ and $r$ are the radii and $d$ is the distance between the centres.
To find the intersections you need to plug the known value of $A$ into that formula and solve the resulting equation for $d$. But there is no analytical formula for that, you must find $d$ by some numerical method.