Both Points Outside A Circle

Steps

1. Construct a circle for circular inversion. Choose the point A as its centre and the distance between the assigned points (the line segment AB) as its radius.
2. Invert the assigned circle into the newly constructed one with the help of circular inversion.
3. After constructing the new circle with c. inversion, construct its tangents which pass through the point B.
4. Construct two circles with the circular inversion of tangents of the circle constructed in the first step. The two circles, k1 and k2, are the solution.

Steps

0. We have a circle k with center S and points A, B
1. Draw a line f passing through points A and B and the perpendicular bisector of segment AB, g
2. Draw a circle c with its center on g having an arbitrary radius
3. Mark the intersections of c and k as E and F, and label the line EF as h
4. Mark the intersection of f and h as G
5. From G, draw tangents to k, which we label as i and j
6. From the intersections of i and j with k, labeled as H and I, respectively, draw lines HS, labeled as l, and IS, labeled as m
7. Mark the intersection of g with m as J and the intersection of g with l as K
8. Construct the finalised solution by creating k1, with center at J passing through A, and k2, with center at K passing through A