On the cubic equation with its Siebeck–Marden–Northshield triangle and the quartic equation with its tetrahedron

Prodanov, 2020, On the determination of the number of positive and negative polynomial zeros and their isolation, Open Math. (de Gruyter), 18, 1387, 10.1515/math-2020-0079 Prodanov, 2021, Classification of the real roots of the quartic equation and their pythagorean tunes, Int. J. Appl. Comput. Math. (Springer), 7, 218, 10.1007/s40819-021-01152-w Prodanov, 2022, Isolation intervals of the real roots of the parametric cubic equation and improved complete root classification, Adv. Theory Simul. (Wiley), 10.1002/adts.202100638 Prodanov, 2022, A method for locating the real roots of the symbolic quintic equation using quadratic equations, Adv. Theory Simul. (Wiley), 10.1002/adts.202200011 Rees, 1922, Graphical discussion of the roots of a quartic equation, Amer. Math. Monthly, 29, 51, 10.1080/00029890.1922.11986100 Viète, 1646 Descartes, 1637 Nickalls, 2006, Viète, descartes and the cubic equation, Math. Gaz., 90, 203, 10.1017/S0025557200179598 Northshield, 2013, Geometry of cubic polynomials, Math. Mag., 86, 136, 10.4169/math.mag.86.2.136 Siebeck, 1864, Ueber eine neue analytische behandlungweise der brennpunkte, J. Reine Angew. Math., 64, 175 Marden, 1966 Marden, 1945, A note on the zeros of the sections of a partial fraction, Bull. Amer. Math. Soc., 51, 935, 10.1090/S0002-9904-1945-08470-5 Kalman, 2008, An elementary proof of Marden’s theorem, Amer. Math. Monthly, 115, 330, 10.1080/00029890.2008.11920532 Badertscher, 2014, A simple direct proof of Marden’s theorem, Amer. Math. Monthly, 121, 547, 10.4169/amer.math.monthly.121.06.547 Steiner, 1828, Géométrie pure. Développment d’une série de théorèmes relatifs aux sections coniques, Ann. Math. Pures Appl., 19, 37 Prodanov, 2022, The Siebeck–Marden–Northshield theorem and the real roots of the symbolic cubic equation, Resultate der Mathematik (Birkhäuser), 77, 126, 10.1007/s00025-022-01667-8 Chalkley, 1975, Quartic equations and tetrahedral symmetries, Math. Mag., 48, 211, 10.1080/0025570X.1975.11976489 Nickalls, 2012, The quartic equation: Alignment with an equivalent tetrahedron, Math. Gaz., 96, 49, 10.1017/S0025557200003958 Nickalls, 2009, The quartic equation: Invariants and Euler’s solution revealed, Math. Gaz., 93, 66, 10.1017/S0025557200184190 Cayley, 1861, A discussion on the Sturmian constants for cubic and quartic equations, Q. J. Pure Appl. Math., IV, 7 Sturm, 1829, Bulletin Général et Universel des Annonces et des Nouvelles Scientifiques, Bull. Sci. Férussac, 11, 419 Petrovich, 1909, Une Classe Remarquable de Séries Entiérs Liang, 2008, The complete root classification of a parametric polynomial on an interval Yang, 1996, A complete discrimination system for polynomials, Sci. China (Series E), 39, 628