Marden's theorem

Given any ${\displaystyle a,b\in \mathbb {C} }$ with ${\displaystyle a\neq 0}$, define ${\displaystyle g(z)=f(az+b)}$, then ${\displaystyle g'(z)=af'(az+b)}$. Thus, we have $${\displaystyle g^{-1}(0)=(f^{-1}(0)-b)/a}$$

and similarly for ${\displaystyle g'}$ and ${\displaystyle f'}$. In other words, by a linear change of variables, we may perform arbitrary translation, rotation, and scaling on the roots of ${\displaystyle f}$ and ${\displaystyle f'}$.

Thus, WLOG, we let the Steiner inellipse's focal points be on the real axis, at ${\displaystyle \pm c}$, where ${\displaystyle c}$ is the focal length. Let ${\displaystyle a,b}$ be the long and short semiaxis lengths, so that ${\displaystyle c={\sqrt {a^{2}-b^{2}}}}$.

Let the three roots of ${\displaystyle f}$ be ${\displaystyle z_{j}:=x_{j}+y_{j}i}$ for ${\displaystyle j=0,1,2}$.

Horizontally stretch the complex plane so that the Steiner inellipse becomes a circle of radius ${\displaystyle b}$. This would transform the triangle into an equilateral triangle, with vertices ${\displaystyle \zeta _{j}={\frac {b}{a}}x_{j}+y_{j}i}$.

By geometry of the equilateral triangle, ${\displaystyle \sum _{j}\zeta _{j}=0}$, we have ${\displaystyle \sum _{j}z_{j}=0}$, thus ${\displaystyle f(z)=z^{3}+z\sum _{j}z_{j}z_{j+1}-z_{0}z_{1}z_{2}}$ by Vieta's formulas (for notational cleanness, we "loop back" the indices, that is, ${\displaystyle z_{3}=z_{0}}$.). Now it remains to show that ${\displaystyle 3c^{2}+\sum _{j}z_{j}z_{j+1}=0}$,

Since ${\displaystyle 0=\left(\sum _{j}z_{j}\right)^{2}=\sum _{j}z_{j}^{2}+2\sum _{j}z_{j}z_{j+1}}$, it remains to show ${\displaystyle \sum _{j}z_{j}^{2}=6c^{2}}$, that is, it remains to show

$${\displaystyle \sum _{j}x_{j}y_{j}=0;\quad \sum _{j}x_{j}^{2}-y_{j}^{2}=6(a^{2}-b^{2})}$$

By the geometry of the equilateral triangle, we have ${\displaystyle \sum _{j}\zeta _{j}^{2}=0}$, and ${\displaystyle |\zeta _{j}|=2b}$ for each ${\displaystyle j}$, which implies

$${\displaystyle \sum _{j}{\frac {2b}{a}}x_{j}y_{j}=0;\quad \sum _{j}{\frac {b^{2}}{a^{2}}}x_{j}^{2}-y_{j}^{2}=0;\quad \sum _{j}{\frac {b^{2}}{a^{2}}}x_{j}^{2}+y_{j}^{2}=12b^{2}}$$

which yields the desired equalities.