When reading about the properties of circles it would seem like a natural step to study (by study I mean, reading about it and pretending to, or at least, hoping to understand) the problem of Apollonius.
What it is, is that for any three given circles of arbitrary size and position, there are a set number of ways to draw a fourth circle sharing tangent points with the first three.
In fact, there are exactly eight different solutions to this problem given that the original circles are not mutually tangent. In that case there are only two solutions (actually five, counting the original circles).
I quickly realized that my mathematical prowess would be no match for this kind of problem so I resolved to another way. Not as nobel but very effective all the same: Given enough people with programming skills, an internet connection and some sense of philantropy someone is bound to post some code that I could steal, erh, borrow.
True indeed, turns out there are lots (by lots I mean virtually everyone) of people infinitely better than me at this stuff. This should probably be the time to give credit to whoever I stole/borrowed it from had I not forgotten where I found it. Sorry!
Anyway, I made this script allowing me to draw these Apollonian Circles which could, some day, turn out to be part of a very nice and powerful set of geometric drawing tools.