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

1. 
 Start with the given circles.
First case:
the circles overlap; they intersect at two points.


2. 
 Draw the connecting line between the centers of the circles.


3. 
 Construct two lines perpendicular to line 2, each passing through one of the centers.


4. 
 Draw the two connecting lines between the intersections of lines 3 with the circles, each at one side of line 2. This is equivalent to the construction of parallel lines at a certain distance.


5. 
 Lines 4 are the lines to be constructed: they both are tangent to both circles. In this case, there are two such lines.


6. 
 Second case:
the circles lie outside of each other, but have one tangent point in common.


7. 
 Draw the connecting line between the centers of the circles.


8. 
 Construct a line perpendicular to line 7, passing through the common tangent point.


9. 
 Construct two lines perpendicular to line 7, each passing through one of the centers.


10. 
 Construct two parallel lines by connecting the intersections of lines 9 with the circles, each at one side of line 7.


11. 
 Lines 8 and 10 are the lines to be constructed: they are tangent to both circles. In this case, there are three such lines.


12. 
 Third case:
the circles lie completely outside of each other.


13. 
 Draw the connecting line between the centers.


14. 
 Construct two lines perpendicular to line 13, each passing through one of the centers of the circles.


15. 
 Construct two parallel lines by connecting the intersections of lines 14 with the circles.


16. 
 Draw the connecting line between the intersection of the first circle with the corresponding line 14, and the intersection of the second circle with the other line 14, at different sides of line 13.


17. 
 Construct two lines, from the intersection of lines 13 and 16, tangent to one circle, and extend these lines upto the other circle.


18. 
 Lines 15 and 17 are the lines to be constructed: they are tangent to both circles. In this case, there are four such lines.

