Calculating diameter of neighboring circles

Question

I have one circle C1 with known diameter d1. That circle is surrounded by N circles (C2) of unknown diameter d2 where each of C2 circles touches two neighboring C2 circles and C1 circle in one point. Have to find d2.

Something like this...

So, what have I tried so far...

I'm not mathematician and to be honest, I haven't been in touch with any by basic everyday math for almost 20 years. If I remember well from school, solution to this problem falls under trigonometry. So I tried to use triangles in many different ways and finally ended up with something like this...

I could solve this by basic trigonometric formulas if any of the sides of triangle is known, but I have only partial data and some relations. So this is my best shot.

I don't know if this is the right way to solve it or do I have enough of known values to get desired result but that's all I have this far. Maybe my approach seems apsurd for most of you doing stuff like this on daily basis, so don't blame me.

Anyway, eneded up here because I'm trying to solve practical problem using math so guess that place where mathematicians are around is good to start with. I couldn't even search the web for solutions just because I really don't know what to look for or technical terms to search for. Hope someone here may help me with formula to get d2 from first figure when N and d1 are known.

Hope I provided enough of details to explain my problem because I really don't know any other way to do this.

Answer 1

You are doing it right. Just use the Law of Cosines on $ABC$ as follows:

$$\cos\left(\frac{360}{N}\right)= \frac{2(r_1+r_2)^2 -(2r_2)^2}{2(r_1+r_2)^2}$$

Answer 2

Of course, this is a simple case, which can be easily solved by basic geometry, but it's also good to know that it's a special case of the Steiner's Chain of $n$ circles.

Given two circles with radii $r<R$, the distance between their centers $d$, in the Steiner's Chain of $n$ circles the mentioned parameters must agree with this equation: \begin{align} d^2&=(R-r)^2-4Rr\tan^2\tfrac\pi{n} \tag{1}\label{1} . \end{align}

For the given special case we have known radius of the inner circle $r$ and distance $d=0$, hence, the radius $R$ of the outer circle can be found as

\begin{align} R&=r\cdot\left(\frac{1+\sin\tfrac\pi{n}}{\cos\tfrac\pi{n}} \right)^2 , \end{align}

hence, the radius, common for all of $n$ circles,

\begin{align} r_n&=\tfrac12\,(R-r) =\frac{r\,\sin\tfrac\pi{n}}{1-\sin\tfrac\pi{n}} . \end{align}

Answer 3

Just later found another solution in case someone needs it by using similarity of two isosceles triangles and proportion.

Here we have two similar triangles ABC and AB1C1 and after calculating size of B1C1 we can easily use proportions to get r2.

I'd like that I figured that out before I posted this question. Sometimes solution is closer than we think.

Answered own question just in case someone with similar problem may need another approach.