Chapter 14.5, Problem 85E

Descartes’ four-circle solution Consider the four mutually tangent circles shown in the figure that have radii a, b, c, and d, and curvatures A = 1/a, B = 1/b, C = 1/c, and D = 1/d. Prove Descartes’ result (1643) that

${(A+B+C+D)}^2=2(A^2+B^2+C^2+D^2).$