Lill's Construction for a Depressed Cubic Polynomial

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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).


Contributed by: Izidor Hafner (September 2017)
Open content licensed under CC BY-NC-SA



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].


[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.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.