Astonishingly, no mathematician ever could give a "Mr. Foobar invented this" whenever I came up with this construction, although it is very elementary.
Given are 3 circles C1,C2,C3 (avoid degenerate configurations for now). Let L be the geometric locus of the centers of all circles C which intersect C1, C2 and C3 under the same angle @ (which may be non-real - doesn't hurt!)
Clearly the radical center (@=90°) and the all-outer/inner Apollonius center (@=0/180°) lie on L, and some analytic geometry immediately shows L is a straight line.
Bonus Track (only if you have too much time): Calculate @ for the Gergonne point when L is the Soddy line of C1, C2, C3. A most surprising result awaits. (Purely geometric proof, anyone?)
Edit: (Added from comments)
Here's an image:
The dotted circle is for @=120° (of course everything is drawn only approximate!)