I was looking at this problem:
https://en.wikipedia.org/wiki/Circle_packing_in_a_circle which roughly asks what is the optimal packing of n circles of the same area $q$ into a larger circle. For simplicity I let the area of the larger circle be 1. Moreover what is the value of $nq$.
I was focusing $n=3,4$ (The case of 4 appeared in a Math GRE exam) but after chewing on it for some time it was not clear to me how to prove optimality,
It's easy to reason for $n=2$ that each circle must touch the boundary (if it didn't you can always shift and expand it) and from there the diameter argument follows.
For $n=3$ We have one circle touching the boundary, but now it's not obvious to me how to solve this without resorting to some non-linear optimization problem. I think I'm missing some geometric intuition on how to prove that they need to be symmetrically distributed.