The full question is as follows:
Suppose $X, Y, Z$ are three different, circles of equal radius which are mutually tangent. Let circle $A$ be the circle tangent to $X, Y$, and $Z$ inside the gap between them, and let circle $B$ be the circle tangent to $X, Y$, and $Z$ that surrounds them. Find the ratio of the radius of $B$ to the radius of $A$ in the form $a + b \sqrt{c}$ where $a, b, c$ are integers.
To start I created an illustration of the problem.
Note that the illustration may not be completely to scale, I created with with shapes on Microsoft Word
I called $r_o$ the radius of circles $X,Y,Z$. I also labeled $r_b$ the radius of circle $B$ and $r_a$ the radius of circle $A$.(I withheld from including the last two in the illustration because I felt the picture would become hard to navigate)
From here I noticed that $$r_b = 2 r_o + r_a$$ So the ratio of radius $B$ to radius $A$ is $$\frac{r_b}{r_a} = 2\frac{r_o}{r_a} + 1$$
My question arises when solving for $r_o / r_a$
From the sketch it looks like the centers of circles $X,Y,Z$ form an equilateral triangle, with side length $2r_o$. Then from that triangle I created a smaller isosceles triangle with base length $2r_o$ and angles $30^\circ$, $30^\circ$, $120^\circ$ . (represented by the blue dashed lines in the illustration)
Are my assumptions valid and/or correct?
After my assumptions I used the Law of Sines to find $r_o + r_a$ in terms of $r_o$ thus using that to find $r_o / r_a$ , and my final answer for the whole problem is $7 + 4\sqrt{3}$