The Siebeck–Marden–Northshield Theorem and the Real Roots of the Symbolic Cubic Equation

Abstract

The isolation intervals of the real roots of the symbolic monic cubic polynomial x^3 + a x^2 + b x + c are determined, in terms of the coefficients of the polynomial, by solving the Siebeck–Marden–Northshield triangle—the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial.