So do not even suggest this question may be a rip-off of https://puzzling.stackexchange.com/q/109502/73836

From a pile of equal size perfectly round coins take eigtheen and make a perfect ring. Show that you can fit at least sixteen more coins inside the ring.

[![Not a solution because twelve is not sixteen.][1]][1]

*Not a solution because twelve is not sixteen.*

Please, do so without resorting to a computer, without "lateral thinking" and without referencing http://packomania.com/ or friends.

Note: The intended solution is pretty symmetric. The task is mostly about proving that it works.


  [1]: https://i.sstatic.net/o0Ao2.png