Lill's Construction for a Depressed Cubic Polynomial

This Demonstration shows a graphic check of Lill's construction of a polynomial of the form where .
Consider the trigonometric identity for in the form
.
Substitute and to get the polynomial.
The equation
has the solutions
,
,
.
By Lill's construction,
,
so given , we must find such that (the length of the thick red segment is 0).

SNAPSHOTS

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DETAILS

Since we are mainly interested in zeros of the polynomial , it looks like this Demonstration is less general than the so-called depressed equation . But Vieta showed that in the irreducible case, using the substitution gives the equation [2, p. 133].
References
[1] D. Kurepa, Higher Algebra, Book 2 (in Croatian), Zagreb: Skolska knjiga, 1965 pp. 1072–1074.
[2] G. E. Martin, Geometric Constructions, New York: Springer, 1998.
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