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 Illustration of problem

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 $7 + 4\sqrt{3}$

  • 1
    $\begingroup$ pballew.net/soddy.html is useful. $\endgroup$ – Narasimham Feb 25 '17 at 20:54
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    $\begingroup$ en.wikipedia.org/wiki/Descartes'_theorem too. $\endgroup$ – Jack D'Aurizio Feb 25 '17 at 21:04
  • $\begingroup$ Your assumptions are correct and all seems fine to me. $\endgroup$ – Emilio Novati Feb 25 '17 at 21:10
  • $\begingroup$ Seems like Descartes Circle Theorem would have been a much faster way to figure this problem out. Will try to learn this so I know for next time. $\endgroup$ – WaveX Feb 25 '17 at 21:17

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