
How to construct any circumscribed regular polygon, given the corresponding inscribed regular polygon?

1. 
 Start with the construction of the circumscribed circle of the given regular polygon, if not yet already done. For instance, use the construction of the "threepoints" circle to do so.


2. 
 Select two adjacent angular points of the given polygon, and construct, if not yet done so, the corresponding rays from the center of circle 1 to these points.


3. 
 Copy circle 1 two times, to the selected points 2 (by constructing two circles centered at the points 2, both passing through the center of circle 1).


4. 
 Draw the connecting line between both intersections of circles 3. This line is the bisector of the angle between rays 2.


5. 
 Extend one of rays 2 up to the opposite side of the corresponding circle 3.


6. 
 Construct two arcs, one concentric to circle 1, the other centered at the intersection of line 5 and the corresponding circle 3, each passing through the other one's center, as shown. Both arcs must intersect at both sides.


7. 
 Draw the connecting line between both intersections of arcs 6. This line is perpendicular to the corresponding ray 2.


8. 
 Construct a circle concentric to circle 1, passing through the intersection of lines 4 and 7. The circumscribed polygon to be constructed is the inscribed polygon of this circle.


9. 
 Construct a circle centered at the same intersection of lines 4 and 7, passing through the other intersection of line 7 and circle 8. Construct two circles centered at the intersections of the last one(s) with circle 8, passing through its center(s), as shown. Repeat this common construction of the inscribed polygon as many times as needed, all around circle 8.


10. 
 Draw the connecting lines between all adjacent intersections of circle 8 and circles 9.
These lines together make up the regular circumscribed polygon to be constructed.

