What Size Circle Fits Between Three Touching Circles?

[klawlor419]

Hi -- Thinking about the problem, where I have three circles in a closest possible packing inside an equilateral triangle. So two circles on the floor, adjacent to each other touching and a third circle placed on top so that the distances off their centers from each other are all 2R, R=Radius.

What I am want to know is the size of the circle that would maximally fill in the space between the three larger circles. Any ideas? I'm thinking calculus min or maximization problem.

Any references, ideas or pointers appreciated.

 

[Simon Bridge]

Sketch it out as coordinate geometry:
If you have already solved for the radius of the three circles - then these will define three points on the inner circle and thus the circle itself can be found.

i.e. the inner circle you are finding is centered at (0,0), and the upper circle is centered at (0,y), and has radius r, then the radius of the inner circle is y-r

 

[klawlor419]

Simon -- Thanks for the response. So I got that the radius of the small inner circle turns out to be, r=2Sqrt[3]-3. I'm not sure if this is correct but the number seems to make sense. Turns out ~.46 units as opposed to the outer spheres which I assumed to be 1 unit radii.

ratio(outer:inner)=(2/3)Sqrt[3]-1

Can this be confirmed? Thanks again

 

[Whovian]

You might want to look at http://en.wikipedia.org/wiki/Descartes'_theorem]Descartes'[/PLAIN] Theorem. Or you could draw a bunch of line segments (such as those connecting the centres of the circles and the radii tangent to the triangle) and apply the Law of Cosines a bunch of times to find it after assuming that the system has 3-fold rotational symmetry, which works similarly well.

 

[klawlor419]

Whovian -- Thanks. Who would have thought such a seemingly simple problem, has such a deep history. Thanks for the pointer.

 

[Simon Bridge]

... but without seeing your working, we cannot really comment.