Let E be the ellipse of maximum area that can be inscribed in a triangle of side lengths 1, √3, 2.
Find the distance between the foci of the ellipse E.
The three values 0, i, and sqrt[3] plotted on the complex plane form a triangle with dimensions 1, sqrt[3], 2. The equation f(x) = x^3 - (i+sqrt[3])x^2 + (i*sqrt[3])x has the three complex values as roots.
Then apply
Marden's Theorem, which states that the roots f'(x), the derivative of f(x), are the locations of the foci of the Steiner inellipse. In this case f'(x) = 3x^2 - (2i+2*sqrt[3])x + i*sqrt[3], whose roots are ((i+sqrt[3]) +/- sqrt[2-i*sqrt[3]]/3.
The distance is the absolute value of the difference of the two roots, which simplifies to abs((2/3)*sqrt[2-i*sqrt[3]]) which does simplify to (2/3)*7^(1/4).
Complex numbers
