"Fake" Construction

Fake” construction mainly includes methods such as marking a straight edge or not approved use of compass while constructing certain diagrams. A famous example is trisecting an angle.

Since it is impossible to create the length the same as the radius of the circle, this is not an acceptable construction. However, with these extended rules we can do many more constructions which are useful, like a heptagon. What else can you construct with these broadened rules?