baragar.faculty.unlv.edu/papers/TwiceNotch.pdf

Arthur Baragar proved the equivalence of neusis and conchoid-assisted constructions, and that all complex numbers constructible by neusis/conchoid lie on a finite tower of field extensions of degrees 2, 3, 5, and 6.

If you draw this conchoid on a paper, such that one of the curves has a loop, you can draw a circle that intersects the conchoid at six distinct points, which means their coordinates can not all be obtained with a quartic (let alone quadratic) equations.

As Oscar Lanzi pointed out, Snyder and Benjamin proved that the regular 11-gon is constructible by this method (and consequently any regular 11n-gon, where n is the number of sides of a regular polygon constructible by compass, straightedge, and hyperbola)

baragar.faculty.unlv.edu/papers/TwiceNotch.pdf

Arthur Baragar proved the equivalence of neusis and conchoid-assisted constructions, and that all complex numbers constructible by neusis/conchoid lie on a finite tower of field extensions of degrees 2, 3, 5, and 6.

If you draw this conchoid on a paper, such that one of the curves has a loop, you can draw a circle that intersects the conchoid at six distinct points, which means their coordinates can not all be obtained with a quartic (let alone quadratic) equations.

As Oscar Lanzi pointed out, Snyder and Benjamin proved that the regular 11-gon is constructible by this method (and consequently any regular 11n-gon, where n is the number of sides of a regular polygon constructible by compass, straightedge, and hyperbola)