
How to copy a circle according to the "strict" rulerandcompass method?

1. 


Start with the circle to be copied (circle 1), and determine the destination point it is to be copied to (center of the circle to be constructed).
There are several possibilities for this destination point. If it is the center of the original circle (for whatever reason!), just redo the construction of it.
If it lies on the circle itself (above), then the construction is trivial: construct a circle with its center at the destination point, passing through the center of the original circle.




If neither of both is the case, the destination point can lie outside (left) or inside the circle (right). The construction steps are essentially the same, but look somewhat different.


2. 


Draw the connecting line between the center of circle 1 (the original circle) and the destination point.


3. 


Construct a circle centered at the destination point, passing through the center of circle 1.


4. 


Construct a circle concentric to circle 1, passing through the destination point.


5. 


Draw the connecting line between the center of circle 1 and one of the intersections of circles 3 and 4. Extend this line, if necessary, up to circle 1 (if it is not yet intersecting it). Notice, that circles 3 and 4 make up a socalled "vesica pisces" construction (two equal circles passing through each other's center), and the angle between lines 2 and 5 is 60°.


6. 


Construct a circle centered at the intersection of line 5 and circle 1, passing through the center of circle 1.


7. 


Extend line 5 up to the opposite side of circle 6 (if necessary!).


8. 


Extend line 2 up to the opposite side of circle 3.


9. 


Draw the connecting line between the intersection of circle 6 and line 7 (or line 5 if not extended) and the intersection of circle 3 and line 8.


10. 


Construct a circle centered at the intersection of lines 8 and 9, passing through the destination point.


11. 


Draw the connecting line between the destination point and the intersection of circles 3 and 10 at the same side as line 5, as shown. Extend this line up to line 9 if necessary (if not yet intersecting it). Again, circles 3 and 10 form a "vesica pisces", and the angle between lines 8 and 11 is 60°.


12. 


Finally, construct a circle concentric to circle 3, passing through the intersection of lines 9 and 11. This is the circle to be constructed: a copy of circle 1 centered at the destination point.
This construction only used drawing a line through two points, and constructing a circle centered at a given or constructed point, passing through a (different) given or constructed point. This is in accordance with the "strict" rulerandcompass method.


13. 


Why are circles 1 and 12 equal (have equal radii)? This will be made clear with the assistance of some helplines.
Angle p = 60° (both). Therefore, AB and DE are parallel lines. Since b = c (radii of circle 3) and d = e (radii of circle 6), lines BC and FD are also parallel. That means, that angle r equals angle q (r = q). Within triangle CDE, the sum of angles p, q and s equals 180°, or: s = 180°  p  q. At point D, the sum of angles p, r and t also equals 180° (straight line!) and since r = q, t = 180°  p  q. Consequently, t = s. That makes triangles AFD and DEC to be equal (p = p, b = c and t = s). The result is, that x = d; line d is the radius of circle 1 and line x of circle 12, so, the radius of circle 12 is equal to the radius of circle 1.

