The incenter of a regular hexagon would be found by constructing the angle bisectors of each vertex and finding their intersection point. This would not involve circles.
We can see from the diagram that the hexagon is not circumscribed about the circle.
Construction of a regular hexagon inscribed in a circle involves creating 6 arcs the length of the radius of the circle; we would not draw this many circles.
This leaves the circumcenter of a regular hexagon as our answer.
