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 **P** to be a point in the intersectionof two circumcribed circles, and also in the Voronoicell of a third disc.The two discs are close enough that thepoint exactly half-way between them will be in the dead region where thetriple point cannot be located.The triple point must therefore lie between **P** and this half-way point,and also in both hexagonally circumscribed circles.But each of the vertex angles at the capped discsfrom this triple point must then be at least**120**^{o}. This can only happen ifthis angle is exactly **120**^{o} and the three discs are mutually touching. In the diagram to the right, this means that **the pointsin the yellow rhombus are never in the Voronoi cellof the third disc.** | The triple point never lies inside the hexagonally circumscribed circles. As a consequence, points in the yellow rhombus never lie inthe Voronoi cell of a third disc.Disc three is *live*. |