One construction for a 17-sided polygon is shown below.
Construct a circle with diameter and radius OC perpendicular to AOB
Construct OD = OC/4
Construct DA
Bisect angle ODA twice to construct angle ODE (= ODA/4)
Construct angle EDF = 45 degrees
Construct a semicircle with AF as diameter, intersecting OC at G
Using E as center and EG as radius, construct a semicircle intersecting AB at H and K.
Construct HL and KM, both perpendicular to AB
Angle LOM = 720/17. Bisect LOM to get 360/17.
Prove this.