This sketch was inspired by a picture Patrick Honner posted of a student diagram.
http://mrhonner.com/2012/03/08/worlds-most-complicated-geometry-diagram/
I got stumped trying to construct a circle exterior to the three circles and tangent to all three. Constructing the three circles tangent to all three sides was interesting in itself. It's an example of an ancient problem set: construct a circle tangent to these three objects (points, lines, circles, etc.) that comes from circles being determined by three points.
Since I was stumped, I made this sketch to gather data, look for patterns and make conjectures. What determines where those tangent points need to be?
This is connected to Apollonius Problem: http://mathworld.wolfram.com/ApolloniusCircle.html
(which is usually solved algebraically) See also Alexander Bogolmolny's applet at Cut the Knot's: http://www.cut-the-knot.org/Curriculum/Geometry/Apollonius.shtml