I am currently working on the following problem (see also the example drawing): Suppose I have, for example, a hexagonal circle packing or a square arrangement. And we randomly place a point in a circle in this arrangement. Is there a formula to (relatively) easily calculate what the average distance of a random point to the second, third, fourth, ... closest circle centre within the arrangement is? In addition, I am also interested in the problem if there is no infinite plane, but if, for example, the 7 circles shown in the example drawing would be the only circles in my circle packing arrangement.
I'm not a circle packing expert at all, but I have the feeling that such a problem might have been investigated before?
Alternatively, I could calculate it with a computer program. However, this would of course be quite inefficient? Nevertheless, I would also be grateful if someone knows code that already calculates this problem.
Hint: The issue can be simplified by looking at the distribution of distances from variable point $M$ to only three points which are the vertices of an equilateral triangle like $C,A,E$. Moreover, one can consider at first that $M$ can be anywhere in triangle $CAE$, constraining it in a second step to stay inside the circle centered in $G$ which is the incircle of triangle $CAE$.
The distribution of distances can be treated using barycentric coordinates. Having this distribution, it is easy to obtain its mean.
Remark: the distribution of distances with respect to the "complementary" triangle $BDF$ is clearly identical to the distribution with respect to triangle $CAE$.
If I have some time in the coming days, I will attempt to make the calculations.