Circles With No Intersects Outside Each Other, Point Lies Outside Both Circles

Steps

1. Draw a circle centered at point A and proceed with a circle inversion.
2. Invert all objects in the problem through the circle inversion from step 1. This leaves point A at infinity.
3. Draw the common tangents of the circles. They meet the point at infinity.
4. We invert the tangents from step 3 through circle inversion from step 1.
5. 4 solutions arise.

Steps

0. The set of all centres of circles tangent to a circle and passing through a point outside the circle is a hyperbola. The foci of this hyperbola are the given point and the center of the given circle. We will look for a pair of hyperbolas, each for a pair of the given point and one of the two circles.
1. To construct both hyperbolas, we still need to find an arbitrary point on each hyperbola. We can always find one such point at the junction of the given point and the center of the circle.
2. At the center of the line segment formed by the given point and the intersection of the line with the circle lies a point of the hyperbola.
3. From knowing the foci and one of the points of the hyperbolas, the two hyperbolas can be plotted.
4. The centers of the solution circles lie on the intersections of the hyperbolas.
5. The problem has a total of four solutions.