Geometry: dividing a line and inscribing a hexagon
Compass constructions dividing a segment into 2, 3, and 6 parts and building a regular hexagon
Three compass constructions fill the sheet. The first bisects line a b (and divides it into three) using two equal circles centered on the segment, their intersections giving the perpendicular bisector. The second adds smaller concentric circles to divide line c d into six parts and into three, and the third inscribes a regular hexagon in a circle by stepping the radius around the circumference to fix chord e f as the side length.
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Dividing line a b into two and three parts
Two circles of equal radius are drawn with centers on the segment, each passing through one endpoint of line a b. Their intersections determine the perpendicular bisector, dividing the line into equal parts with a single compass opening.
Dividing line c d into six parts
To the previous construction two smaller circles are added, each concentric with a larger one, so that line c d is divided into six equal parts and into three.
Inscribing a regular hexagon on e f
An arc of the circle's own radius is stepped through center and circumference to fix point e, and again to fix point f. The chord e f gives the length of each of the six sides of the hexagon inscribed in the circle.