Two Separate Circles with Common Tangent on Other Side

Steps

1. Choose the circle of inversion with its center at the tangency point of the given circle and the line.
2. Apply circular inversion to the given objects. The circle passing through the center of the inversion transforms into a straight line. The given line is self-inverse.
3. Find the circle tangent to the transformed objects. To locate its center, use loci of points defined by specific geometric properties.
4. The solution that passes through the tangency point (the center of the inversion circle) is transformed into a tangent to the image of the circle, parallel to the given line.
5. Invert the found images of the solution circles back to obtain the original solutions.
6. The problem has two solutions.

Steps

1. The centers of all circles that are tangent to the first given circle and the given line at their point of tangency lie on the perpendicular to the line passing through the common tangency point.
2. The centers of all circles that are tangent to the first given circle and the line at a point other than their point of tangency lie on a parabola. The focus of this parabola is the center of the circle, and the directrix is a line parallel to the given line. The distance between the circle and the directrix is equal to the radius of the circle.
3. The centers of all circles that are tangent to the second given circle and the given line at their point of tangency lie on the perpendicular to the line passing through their common tangency point.
4. The centers of all circles that are tangent to the second given circle and the line at a point other than their point of tangency lie on a parabola. The focus is the center of the circle, and the directrix is a line parallel to the given line. The distance between the circle and the directrix is equal to the radius of the circle.
5. Find the intersections of the respective perpendiculars and parabolas. These points are the centers of the solution circles.
6. The problem has two solutions.

Steps

1. The points of tangency of the given circles with the given line will also be the points of tangency with the solution circles. Therefore, the centers of the solution circles will lie on perpendiculars to the line drawn through the points of tangency.
2. To find the solution circles, we use two homotheties in which the solution circles are mapped onto the given circles. The centers of these homotheties are the remaining two points of tangency between the circles. In these homotheties, the given line is mapped to lines tangent to the given circles and parallel to each other.
3. Points of tangency are mapped to points of tangency. The lines passing through them also pass through the centers of the homotheties. These are the intersection points of the lines with the given circles. These points are also points of tangency between the given circles and the solution circles.
4. The centers of the solution circles lie on the perpendicular bisectors of segments defined by the points of tangency.
5. The centers of the solution circles lie at the intersections of the perpendiculars and the bisectors.
6. The problem has two solutions.