
How to construct a line tangent to two given circles of different sizes?

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.

