Nicomedes's Geometric Construction of Principal Cube Root
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This Demonstration shows Nicomedes's method for constructing a cube root using a marked ruler. (Nicomedes was born about 270 BC.) This is actually the real principal cube root. Every nonzero number also has a pair of complex conjugate cube roots.
Contributed by: Izidor Hafner (September 2017)
Open content licensed under CC BY-NC-SA
This Demonstration is based on [1, pp. 128, 129]. The proof from  follows.
Let the line parallel to that passes through intersect at . So . Then, since bisects , . Also, since , we have . With , by two applications of the Pythagorean theorem,
which reduces to the quartic equation
Fortunately, this quartic easily factors as . Since , .
 G. E. Martin, Geometric Constructions, New York: Springer, 1998.