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Proof of Descartes circle formula and its generalization clarified

In his talk"Integral Apollonian disk Packings"Peter Sarnak asked if there is a"proof from the Book"of the Descartes theorem on circles. A candidate for such a proof is presented in this note

In his talk "Integral Apollonian disk Packings" Peter Sarnak asked if there is a "proof from the Book" of the Descartes theorem on circles [Sar]. A candidate for such a proof is presented in this note. Theorem 1 (Descartes formula, 1643 [Ped]). The curvatures a, b, c and d of four pairwise externally tangent disks satisfy (a + b + c + d) 2 = 2 (a 2 + b 2 + c 2 + d 2 ).
A system of such four pair-wise tangent disks is said to form Descartes' configuration.
What follows is a proof of a generalized theorem for disks in general configuration, which will be stated at the end, and from which (1) follows as a special case. This exposition is a succinct and clarified version of the proof in [JK].
Proof: Define a "scalar product" for the disks in the plane as the cosine of the angle under which the circles intersect. In case they do not intersect, we may extend the definition by the Law of Cosines: where d is distance between the centers, and r1, r2 are the corresponding radii of the disks, and ϕ is the angle under which the disks intersect.
There is a map that sends disks (circles) in the plane to space-like unit vectors in M, namely, a disk centered at (x, y) and radius r is mapped to vector where β = 1/r is the curvature, x  = x/r, and y  = y/r are "reduced coordinates.", Now, the inner product (2) becomes Indeed, with simple algebraic manipulations we get: which has manifestly the form of an inner product defined by (3). In particular, the circle vectors are normalized: where G = g -1 is the inverse of g, and F = f -1 is the inverse of f. Next, multiply both sides of the equation by D on the left and D T on the right to arrive at And this is Descartes' theorem generalized! It may be formulated in the following "stand alone" form: (3) Theorem (Generalized disk/circle formula): Four disks in a general configuration in the plane satisfy the following matrix equation Example: In Descartes configuration, Equation (8) turns (9) into: 1 1 1 1 1 1 1 1 1 1 1 1 It is easy to see that f 2 = 4I, hence F = f -1 = (1/4)f. Scaling both sides by 4 we arrive at the extended Descartes' formula for curvatures, positions and co-curvatures (cf. [LMW]): x y x y x y x y The part enclosed by rectangles is Descartes' formula (1).

Remark 1:
The literature often presents Descartes theorem as a statement about circles. It is less confusing to view it as a theorem on configuration of non-overlapping disks. In particular, the "outer circle" in Figure 1(b) represents the boundary of an unbounded disk outside of circle D, for which we assume a negative radius and curvature. The straight lines in Fig. 1(d) represent halfplanes -disks of curvature 0, which are also tangent externally with the other disks.