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?
How to describe foсi of en ellipse inscribed in the triangle thru triangles angles points?
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3What do you mean by "triangles angles points"? – 2011-12-31
1 Answers
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.