Circle-packing question?

Question

You can fit exactly 6 identical circles around the perimeter of a 7th. How can this be shown mathematically?

Answer 1

Tesselate the plane into hexagons, and inscribe circles in the hexagons. Problem solved!

Another way to do it is to show that 3 equal circles touching each other form an equilateral triangle with their centers. From this, show that 6 equilateral triangles can be packed together about a common center. From this, show that 6 circles can be packed around a circle in the center.

Answer 2

You take a regular hexagone and its center. This gives you six equilateral triangles. So you take the seven circles centered at those points and with radius 1/2 the distance between your points and that's it.