The locus line is often called the Chordal of circles A and B.
?
Yeah, me either. I will call it the Power Line, Tangentipede, or Tangent Monster of A and B.
Must consult the manual on Plastic Brontosaurus Algebra.

Circles α, β have equal power at all points on the line. However, if the circles intersect, part of the locus line is a chord. The power at these points is negative, and it is not possible to draw tangents from them. Hence, the locus of points with equal tangents to two circles is all points on the power line whose power is positive (... which are outside both circles).

The construction is partly backward: the radius of β is given by the blue circle, and I make up the perpendicular CF. These are not magic answers. They are tools I use to nose around the structure of a problem. I don't know how other people do it.
Here, we bump into the solution right away by working backward. This is rarely the case. For Malfatti's Problem, for example, the circles in the first worksheet do not obey the constraints: http://www.geogebratube.org/material/show/id/32079. I figured out how to enforce them only when I found the answer to the problem.
It is never necessary to know about an answer in advance, to solve a problem.

_______
Have you noticed that some teachers think that hiding their work from you helps you learn?
I have.
I hate it infinity.
Dear the student: If you have a teacher who does this, use the magic wand whenever possible. When you know how to do something, leave it out, and draw a magic wand instead. If your teacher wants to know how you got the answer, you can compare notes.
I'm happy to show you how I did it. Do you mind showing me how you do it?
Make sure to draw a good wand, though.
You want a quality magic wand.
{End Soapbox}