Well... nobody responds... let me give a proof I like:

Proof: Suppose the radii are . Now instead of circles, imagine three BALLS with radii . Place them in such a way that 1) centers of the balls are on a same plane (call this plane X) and 2) any ball touches the other two. Wrap 2 balls with a cone as in the attachment. For the three balls then we can wrap 3 cones. Ajust the vertex of the cones so that all of them falls on plane X.Notice that now the balls and cones are symmetric about plane X.Let plane Y be the plane that touches the 3 balls on one side of X and plane Z touches the 3 balls on the other side of X. Then the intersection of Y and Z falls on X, too, because of the symmetry. Clearly, A, B, C . Q.E.D.

---------------------------------------------------------------------

The proof can be made cleaner and more rigorous. But we see the point... By escalating the problem to a higher dimension we actually make itEASIERto solve. I have this feeling not just for this particular problem: by engaging ourselves more than a problem seems to require is often a better way to work it out... paradoxically lovely idea