Step 1: Construct a square in a circle (ABCD)
Step 2: Construct a circle in the square
Step 3: Construct an octogon in the circle with two vertices of the octogon on the line that goes through two vertices of the square.
Step 4: Bisect the sides of the octogon and draw lines through the midpoints of the sides of the octogon and the vertices of the octogon.
Step 5: Where the lines cross the outer circle, put points. (There is no need to extend these lines any further).
Step 6: Draw line segments between all of the points on the outer circle.

The resulting polygon (BEFGAHIJDKLMCNOPB) is a regular 16-gon.

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