Find the circle center point that is tangential on 2 other circles that has at least one intersection?

Question

Given circle A and circle B (we got their center points, radiuses and 1 relevant point where they intersect)

we want to draw a circle C with a given radius in a way that it is tangential with the other two circles (there are up to 4 possible locations). How to get the possible centerpoints?

Answer 1

Hint: What is the distance between the centers?

To find (say) $O_3$ (which is internally tangent to both $A$ and $B$), what is $|KO_3|$ and $|O_3M|$?
With that info, determine the possible locations of $O_3$.
Note that there are 2 possible locations. (Why?)

Answer 2

Hint: two circles are tangent at a point $P$ means that $P$ is on the line connecting the two centers.

What does that mean? For example, $M, O_3, N$ are on the same line. Given the radii $R$ and $r$ of the two initial circles, the radius $x$ for the small circle, and the distance $d$ between the original centers, the triangle formed by the three centers has sides $d, R\pm x, r\pm x$. That's a fully determinate problem.