january utc so glad that you liked it thanks for explaining the missing solutions although i now realize that the trick of expanding or shrinking the radii to get a tangency doesn't always work how do you shrink expand a line to keep it in contact with the separating circles as an aside i've heard of sophus lie but i know basically nothing about him or his sphere p on a whim i worked a little more on apollonius' problem today i hope you like it too if you could help me understand the gergonne solution described at mathworld or on the spanish interwiki that'd be great but if you don't have time that's ok too i'm sure that some angel will whisper the answer in my ear if i just stare at it long enough either that or she'll whisper willow you should clean your kitties' litterbox tonight