Intersecting Lines, Intersection Point as Circle Center

Steps

1. The centers of all solution circles lie on the angle bisectors between the given lines. The intersections of these bisectors with the given circle are the points of tangency between the solution circles and the given circle.
2. The tangents to the given circle passing through these points of tangency are also tangents of the solution circles.
3. The tangent and the two given lines form a triangle. Two of the solution circles are the incircle and excircle of this triangle. Their centers lie on the angle bisectors.
4. We have thus found the first two solution circles.
5. Repeat the entire process in the adjacent angle.
6. The remaining solutions can be found in the same way. Alternatively, we can use central symmetry of the remaining four solutions with respect to the center of the given circle.