Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$
We draw a line $d$ through the center of the circle (we can construct it as the intersection of two chords if it's not given) and perpendicular to the given line $g$. $d$ cuts the circle at a two points. Through one of them we construct a line $g'\|g$. $g'$ is parallel to $g$ and tangent to $K$. We may now construct the images of two points on $g'$ under the inversion of center $P$ which preserves $K$. Call the two images $X,Y$. The circle $(PXY)$ is what we are looking for.
Trace a generic circle C tangent in P to g. If R and S are the common points to K and C let T=RS^g. Let T1 and T2 be the point where tangents from T to K touch K.
Let b1 and b2 the angle bisectors of <PTT1 and <PTT2 respectively. The intersections of b1 and b2 with the perpendicular to g in P give us the centers of the two circles wich are solution of the problem.