Packing a Circle with Circles

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.


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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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