Euclid has a magical compass with which he can trisect any angle. Together with a regular compass and a straightedge, can he construct a regular heptagon?
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Gleason's article "Angle Trisection, the Heptagon, and the Triskaidecagon" (also available here) mentions a construction due to Plemelj:
The exact construction for the regular heptagon by angle trisection
Let the triangle ABC with AB=r , AC=3r/2 and BAC=60 deg , has D on AC so that angle ABD is one-third of the angle ABC. Then BD is the side of the regular heptagon inscribed in a circle of r. The hypotenuse of the right triangle with legs 5 and 6 , may be assumed for heptagon side (r=9).
based on Vieta's ideas 1593 - prepared by Ryszard Lesny, 2016