How to describe foсi of en ellipse inscribed in the triangle thru triangles angles points?

Question

I was looking at Marden's theorem and could not help but wonder how foсi of en ellipse inscribed in the triangle can be described thru triangles angles points?

Answer 1

This will not fully answer the question---or maybe it will, since the question is phrased in a way that leaves me uncertain of its intended meaning.

Suppose the three vertices are $z_1,z_2,z_3$, and those are complex numbers.

Let $p(z)=(z-z_1)(z-z_2)(z-z_3)$ be the polynomial whose roots are those vertices. Marden's theorem says the foci of the Steiner inellipse are the zeros of $$ \begin{align} p'(z) & = \frac{d}{dz}(z-z_1)(z-z_2)(z-z_3) \\ \\ & = \frac{d}{dz}(z^3-z^2(z_1+z_2+z_3)+z(z_1z_2+z_1z_3+z_2z_3)-z_1z_2z_3) \\ \\ & = 3z^2- 2z(z_1+z_2+z_3) + (z_1z_2+z_1z_3+z_2z_3). \end{align} $$ Setting this equal to $0$ and solving the quadratic equation, we get $$ \frac{z_1+z_2 + z_3}{3} \pm \frac13\sqrt{z_1^2+z_2^2+z_3^2-z_1z_2-z_1z_3-z_2z_3}. $$ The term before the "$\pm$" is the average of the three vertices, so that's the center of the ellipse.

But what about other ellipses inscribed in the same triangle? The Wikipedia article attributes a result to Linwood that finds ellipses each of whose points of tangency is a rational weighted average of two vertices. How far beyond that we can take the matter may take some further work.