Two Circles With Inner Contact, Line Outside The Circle

Steps

1. We choose the reference circle for the circular inversion. Its center is chosen as the common point of tangency of the two given circles.
2. We apply circular inversion to the given objects. The circles are transformed into a pair of parallel lines, and the line is transformed into a circle.
3. The solution circles are mapped, under inversion, to tangents to the circle that are parallel to the pair of parallel lines.
4. We map the found tangents back using circular inversion.
5. The problem has two solutions.

Steps

1. The solution circles must be tangent to both given circles at their common point of tangency. The set of all centers of circles tangent to both given circles at this point lies on a line passing through the point of tangency and the centers of the given circles.
2. The line that is the common tangent of the given circles at their point of tangency is also a tangent to the solution circles.
3. Since the solution circles must be tangent to both the common tangent and the given line, their centers must lie on the angle bisectors defined by these two lines.
4. The centers of the solution circles lie at the intersections of the angle bisectors with the line connecting the centers of the given circles.
5. The problem has two solutions.

Steps

1. Given two circles tangent at point T and a line that lies outside them.
2. Draw a line h that passes through the centres of the two circles
3. Create an arbitrary point P through which we draw a perpendicular line i to the given line. Find point G - the intersection of lines i & h.
4. Draw a circle e centered at G with radius |GP|. Label the intersections of the circle and the lines i & h with different letters (H, I, J).
5. Draw a line k passing through the points H & P.
6. Draw a line j parallel to the line k passing through the tangent point T.
7. Find the intersection K of line j and the given line. Draw the perpendicular line l that passes through K. The intersection of lines h & l is the center of the first solution.
8. To find the second solution draw a line m that passes through points I & P.
9. Draw a line p that is parallel to line m and also passes through the tangent point T.
10. Find point L - the intersection of lines h & p. Then draw a line q through this point perpendicular to the given line. The intersection of lines h & q is the center of the second solution.