What is the locus of the centre of circles that are tangent to two given circles?
I had no idea how to approach the problem so I considered a special case, namely one in which the two circles were equal. The answer was simple enough due to the symmetry of the situation: the line through their points of intersection.
Then I considered another case, when the two given circles were unequal but tangent to each other. I made a sketch in GeoGebra.
After some trial and error I hit upon a line $CD$ which approximately seemed to be the locus. After some experimentation, I got a feel for what this line was. It somehow bisected the angle formed by the circumferences of the two circles as they converged.
I tried to put this intuition into precise mathematical definition but was not been successful. Trying out the general case also yielded the same result.