Three Circles in a Unit Disk: Three Solutions to an Extremal Problem

We study an extremal problem involving three disks inside the unit disk. Two disks are initially placed so that they are tangent to each other and internally tangent to the unit circle. A third disk of maximum possible radius is then inserted, subject to tangency with both original disks. We prove that the minimum possible sum of the areas of the three disks is 11π/18, attained when their radii are 1/2, 1/2, and 1/3. Surprisingly, the minimizing configuration does not consist of three equal disks, despite the symmetry of the problem. Three different proofs are presented, based on Descartes' circle theorem, geometric inversion, and a parametrization using tangency points.