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| 1. | 
| Start with the given circles. If the smaller circle lies completely inside the larger circle, there are no such lines. First case: the smaller circle lies inside the larger circle, and has a tangent point in common. |
| 2. | 
| Draw a line from the center of the larger circle to the common tangent point. |
| 3. | 
| Construct a line perpendicular to line 2, passing through the tangent point. |
| 4. | 
| Line 3 is the line to be constructed: it is tangent to both circles. |
| 5. | 
| Second case: the smaller circle lies partly inside and partly outside the larger circle; they intersect at two points. |
| 6. | 
| Draw a line from the center of the larger to the center of the smaller circle. |
| 7. | 
| Construct two lines perpendicular to line 6, each passing through one of the two centers. |
| 8. | 
| Draw a line from the intersection of the larger circle and the corresponding line 7 (passing through its center), to the intersection of the smaller circle and the other line 7, both at the same side of line 6. |
| 9. | 
| Extend lines 6 and 8 until they intersect. |
| 10. | 
| Construct a line from the intersection of lines 9, tangent to the larger circle. Notice, that there are two such lines. |
| 11. | 
| Lines 10 are the lines to be constructed: they both are tangent to both circles. In this case, there are two such lines. |
| 12. | 
| Third case: the smaller circle lies outside the larger circle, and has a tangent point in common. This is a combination of the first and second cases. |
| 13. | 
| Draw the connecting line between the centers. |
| 14. | 
| Construct the perpendicular line through the common tangent point. |
| 15. | 
| Construct the perpendicular lines through the centers of both circles. |
| 16. | 
| Draw the connecting line between the corresponding intersections. Extend this line and line 13, until they intersect. |
| 17. | 
| Construct two lines from this intersection, tangent to the larger circle. |
| 18. | 
| Lines 14 and 17 are the lines to be constructed: they are tangent to both circles. In this case, there are three such lines. |
| 19. | 
| Fourth case: the smaller circle lies completely outside the larger circle. |
| 20. | 
| Draw the connecting line between the centers of the two circles. |
| 21. | 
| Construct two lines perpendicular to line 20, each passing through one of the two centers. |
| 22. | 
| Draw the connecting line between the intersections of the circles and the corresponding lines 21, both at the same side of line 20. |
| 23. | 
| Extend lines 20 and 22, until they intersect. |
| 24. | 
| Draw the connecting line between the intersections of the circles and the corresponding lines 21, both at different sides of line 20. |
| 25. | 
| Construct two lines, from the intersection of lines 23, tangent to the larger circle. |
| 26. | 
| Construct two lines, from the intersection of lines 20 and 24, tangent to the larger circle, and extend these lines upto the smaller circle. |
| 27. | 
| Lines 25 and 26 are the lines to be constructed: they are tangent to both circles. In this case, there are four such lines. |
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