Solving Cubic Equations Using Dynamic Geometry

In a right-angled triangle ABC with right angle at point B, base BC, perpendicular AB, perpendiculars BD and EF upon AC, perpendicular DE upon BC, DN upon AB, and finally perpendicular NO upon AC are drawn. When length ON is adjusted by varying angle C, so as to equate it with the constant term of the transformed cubic equation, length BD is equal to the value of one of the real roots of the equation.

Other roots can be determined from the resulting quadratic equation.

Perpendiculars BD and EF upon AC, perpendicular DE upon BC when drawn in a right-angled triangle ABC with right angle at point B, base BC, perpendicular AB, and then these perpendiculars bear a common ratio cos C given by EF/DE = DE/BD = BD/AB = cos C, where C is an angle enclosed by sides CB and CA. These ratios are exploited to solve cubic equations.