Packing a Circle with Circles

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Two circles with a radius initially set to are placed in the unit circle. When you drag the locator, the centers and radii of the circles change. The rule for the iteration is to construct the largest possible circles in the curvilinear triangles.
Contributed by: Hans-Joachim Domke (March 2011)
Open content licensed under CC BY-NC-SA
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The packing shown in this Demonstration is similar to that in "The Circles of Descartes" Demonstration, but uses a different mathematical approach than the Descartes circle theorem to find the solution of the inscribed circle of three mutually tangent circles.
Let the inscribed circle meet the three given circles at
,
, and
. Then the equation
reflects the fact that the point
lies on the circle with center
and radius
. For
and
the equations are similar and for
the equation is
. Because
is tangent to one of the three circles at
,
and hence
.
The coordinates of and the value for
are three unknowns, but there are three equations, so the solutions are well determined. There are two solutions because the equations are quadratic but not all of them are valid.
For the solution circles that are tangent to the unit circle, one of the equations reduces to .
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