Tangent Circles, Line Tangent to One of Them

Steps

1. Choose the circle of inversion with its center at the point of tangency of the two given circles.
2. Apply circular inversion to the given objects. The two circles are mapped to parallel lines, and the line is mapped to a circle.
3. Solve the transformed problem for two parallel lines and a circle. This configuration has three solutions, which can be found using loci of points with defined properties.
4. These are the three images of the solution circles.
5. The fourth solution passes through the tangency point of the given circles, which is the center of the inversion circle. It is therefore mapped to a tangent to the transformed circle that is parallel to the two lines.
6. Invert all found images of the solution circles to obtain the final solutions.
7. The problem has four solutions.

Steps

1. Choose the circle of inversion so that its center is the point of tangency between the given line and the given circle.
2. Apply circular inversion to the given objects. The line remains invariant (self-inverse), and the circle passing through the center of the inversion maps to a line parallel to the original line.
3. Solve the problem for two parallel lines and a circle. This configuration yields three solutions, which can be found using loci of points with defined properties.
4. These are the three images of the solution circles.
5. The fourth solution passes through the tangency point, which is the center of the inversion circle. It is therefore mapped to a tangent to the transformed circle that is parallel to the given line.
6. Invert the found images of the solution circles back to obtain the original solutions.
7. The problem has four solutions.

Steps

1. The centers of circles tangent to the given circle and its tangent line at their common tangency point lie on a line perpendicular to the given line and passing through the point of tangency.
2. The centers of circles tangent to the given circle and its tangent line at a point other than their point of tangency lie on a parabola. The focus of the parabola is the center of the given circle, and the directrix is a line parallel to the given line. The distance between these two lines equals the radius of the given circle.
3. The centers of circles tangent to both given circles at their common tangency point lie on the line connecting the centers of the two circles.
4. The centers of circles that are either externally tangent to both given circles or internally tangent to both lie on a hyperbola. The foci of the hyperbola are the centers of the given circles, and the hyperbola passes through the point of tangency of the circles.
5. The centers of the solution circles are located at the intersections of the identified lines, the parabola, and the hyperbola.
6. The problem has four solutions.