NoteMe saying:
The only way I can think of right now is to find a point that has the same lenght from all points.
triggered something off
firstly I simplified the question. I made the point:
A = (0,1)
B = (1,0)
C = (-1,0)

Now it is obvious what the circle is but it might still help us with more complex questions.

I drew the line y=x
that line gives all the points where A&B are equi-distant. Then all I had to do was find where on that line the distance to all three points are the same. Hmm. maybe I should put on my Maths socks.

I'll have to ponder this a bit longer.