apollonian circles: why are radius and center dual?

Question

This figure suggests the radii and centers (regarded as complex numbers) of the Soddy circles satisfy the same equation: $$ a^2 + b^2 + c^2 + d^2 = \frac{1}{2} (a + b + c + d)^2$$ How can the circle and radius be dual in this particular sangaku problem?

http://dl.dropbox.com/u/17949100/soddy.png

Answer 1

Note that there are two new articles on Apollonian packings, downloadable for free, at BULLETIN

Both bibliographies list four articles, from 2003 to 2006, by Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and Catherine H. Yan. All four titles begin Apollonian circle packings: as mentioned by Hew Wolff