Geodesic
Cubic equations, Newton quadrilaterals, and geometric constructions
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 16 (2024) no. 3, pp. 5-11

Voir la notice de l'article provenant de la source Math-Net.Ru

This article discusses the possibility of constructing with a quadrilateral inscribed in a semicircle a ruler and compass. It shows that the problem of constructing an isosceles triangle from its three bisectors is equivalent to the trisection of an angle. Examples are given of parametric families of equations of the third and sixth degree, for which all roots are expressed through square radicals. A condition is identified under which a sixth-degree polynomial is factorized by third-degree polynomials in canonical form. All the factorizations are valid for polynomials with arbitrary complex coefficients.
Keywords: Newton quadrilaterals, trisection of an angle, cubic equations, solution in square radicals, regular polygons. 
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     author = {N. S. Astapov and N. K. Noland},
     title = {Cubic equations, {Newton} quadrilaterals, and geometric constructions},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
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     url = {http://geodesic.mathdoc.fr/item/VYURM_2024_16_3_a0/}
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N. S. Astapov; N. K. Noland. Cubic equations, Newton quadrilaterals, and geometric constructions. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 16 (2024) no. 3, pp. 5-11. http://geodesic.mathdoc.fr/item/VYURM_2024_16_3_a0/