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Given the image below, how would you work out the position and radius of R?

Arc between two circles:

enter image description here

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    $\begingroup$ You don't have enough info to solve for R. Imagine the bottom circles being physical discs. There is more than one size disc that could be supported between them. $\endgroup$ – turkeyhundt Jul 6 '17 at 23:47
  • $\begingroup$ Please include what you've attempted in solving the question. It also would be helpful if you give a geometric description of the image so those that can't see it can still construct the diagram. (eg. Let $ABC$ be a triangle...) $\endgroup$ – Shuri2060 Jul 6 '17 at 23:48
  • $\begingroup$ Let $\,d\,$ be the distance between the centers of the two circles. Then for any $\,R \ge (d-a-b)/2\,$ you can construct a circle of radius $\,R\,$ (externally) tangent to the two given circles. $\endgroup$ – dxiv Jul 6 '17 at 23:51
  • $\begingroup$ I have been looking for more examples to help with working this out and have found this: Image from this site: joshuanava.biz/engineering-3/tangency.html I'm am more of a programmer than a mathematician so I am struggling to read/work out the solution given. Any help would be truly helpful $\endgroup$ – justachap Jul 7 '17 at 3:09
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According to the given, there will be no solution. Imagine the case that if circle B is placed further apart from circle A, the new required arc length will definitely be longer than the original. Therefore, we have to know D, the distance between two centers.

enter image description here

The example you mentioned is a simple one because R, the radius of the circle-to-be-constructed, is known. In that case, we only need to draw the red arc (centered at A with radius = R + a) and similarly the green arc. Their intersection will yield the point C which is the center of the required circle. In this setting, “the triangle inequality” requires $(R + a) + [R + b] \ge D$. That is, $R \ge (D – a – b)/2$, as mentioned by @dxiv.

If R must be a calculated result, then according to the cosine rule (applied to $\triangle ABC$), one more angle must also be known. For example, if $\alpha$ is known, then

$$(R + a)^2 + D^2 – 2(R + a) D \cos \alpha – (R + b)^2 = 0$$

After simplification, we will get a quadratic equation in R. From which, R can then be found if the discriminant is non-negative.

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