Geometry of Cubic Polynomials

amplitude

axis

Drag the vertices to see how any three real numbers, not all equal, are the projections of the vertices of an equilateral triangle in the plane. For a cubic polynomial

p(x)

with three real roots (not all equal), the inscribed circle of the equilateral triangle that projects onto those roots itself projects to an interval with endpoints equal to the roots of

p'(x)

, the derivative of

p(x).