Intersecting Circles with a Common Tangent

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 circles tangent to the transformed objects. To locate their centers, 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 four solutions.

Steps

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