finding $\angle CB$ or e in a triangle made by four circles with just knowing radiuses of the circles

Question

so is it possible to find $\angle CB$ without having any angles knowing that $\angle AC = r_{2}+r_{3}$ and $\angle AB = r_{2}+r_{1}$ or at least finding the coordinates of point $B$ with knowing point $A$ is $(0, 0)$ and point $C$ is $(0, (r_{2}+r_{3}))$

Edit:
it seems it's impossible to find but can we still find side $e$ if it was four circles?

Answer 1

The easiest way to know about an angle in a triangle, based on the sizes of the edges, is the cosine-rule:

$$A^2 = B^2 + C^2 - 2BC \cos(\alpha)$$

From there, you can derive $cos(\alpha)$, giving you the solution of your problem.

Oh, obviously this only works when all circles are touching each other: in case they don't you need one extra piece of information in order to uniquely describe your situation:

Answer 2

Without any additional information, you cannot know what the angle will be, because there are infinitely many triangles that fit your description. There is nothing in your current description that would preclude the point $B$ from being at almost any position on the circle with center at $(0,0)$ and radius of $r_1+r_2$.