
Packing Pennies in the Plane 

Circumscribe a disc with
a regular hexagon,
and circumscribe the hexagon with a circle.
This gives what I call the hexagonally
circumscribed circle of the original disc.
It is a concentric circle whose radius
is 
We shall need also another property of these circumscribed circles.
Suppose two of them intersect, but that the discs
themselves do not intersect. Then that intersection
can only intersect the Voronoi cell of
a third disc if the three discs are mutually touching
in the configuration of
discs in a hexagonal packing.
For suppose In the diagram to the right, this means that the points in the yellow rhombus are never in the Voronoi cell of the third disc. 
the hexagonally circumscribed circles. As a consequence, points in the yellow rhombus never lie in the Voronoi cell of a third disc. 