Adrian van Roomen, in 1596, used the geometric statement: Loci of centers of circles tangent to two circels are two distinct conic sections. If three circles are defined, then four sets of three conic sections exist, whose common intersections are centers of circles tangent to the three defined circles. The common chords of each set of three conic sections are concurrent in the radical center of the three defined circles.

Use analytic geometry and determine the four linear equations on which pairs of centers of circles tangent to three circles are distributed.