Coxeter has a very simple proof at the beginning of his Introduction to Geometry. It is based on some trigonometry and the fact that the area of a triangle with sides a, b, c can be computed as

pr = (p - a)ra = (p - b)rb = (p - c)rc, where

p = (a + b + c)/2; r is the inradius and ra, rb, rc are exradii of the triangle.

As you may realise the quantities p-a, p-b, p-c are the radii of the Soddy circles.

If you can't get hold of the book (which is a jewel addition to every geometry lover's library) I'll try to find time to put the proof on the web.

Relatively expensive, the book is available from amazon.com:

http://www.amazon.com/Introduction-Geometry-2nd-H-Coxeter/dp/0471504580/sr=1-1/qid=1171807024/ref=pd_bbs_sr_1/102-7772941-9832935?ie=UTF8&s=books