Non computational proof sought

Suppose you seek the smallest circle that can contain 11 non-overlapping unit disks. The solution to this problem is known, but the curious thing is that there are two different arrangements that give exactly the same radius for the circumscribing circle. Here is a picture of them:

http://www.buddenbooks.com/jb/pack/circle/11both.gif

It is far from obvious to me that the circumscribing circles have exactly the same radius. I was able to prove it using Maple with a tedious brute force computation. It seems to me there should be some insightful geometric proof, possibly involving some tessellation of the plane, but I have yet to find one. Can anyone help?

There is more information at this link:

11 circles in a circle

I have a similar question for 13 circles. There is a link about that at the above site.