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Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane.

Consider the ellipse inscribed in the triangle, touching the three mid-point of the three sides of the triangle (why the conic individuated by these three point is an ellipse? is this always true?).

Well, the two focal points of this ellipise, are the roots of $p'$.

How can I prove it?

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  • $\begingroup$ Interesting :) I guess it is the conic which passes through the midpoints and is tangent to the sides of the triangle. The existence can be proved using affine transformations (from an equliateral triangle). The unicity follows from the fact that it passes through three points and is tangent to three lines. Anyway, it is interesting to see the connection with polynomials :) $\endgroup$ – Beni Bogosel Nov 19 '14 at 19:09
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    $\begingroup$ Here is the wikipedia article: en.wikipedia.org/wiki/Steiner_inellipse In the end, it mentions that the above theorem as Marden's theorem: en.wikipedia.org/wiki/Marden's_theorem Here is an elementary proof: dankalman.net/AUhome/pdffiles/mardenAMM.pdf $\endgroup$ – Beni Bogosel Nov 19 '14 at 19:17
  • $\begingroup$ WHOA! Thank you Beni!! :-) $\endgroup$ – Joe Nov 19 '14 at 19:54
  • $\begingroup$ @Beni: I'd say that comment should be an answer. $\endgroup$ – MvG Nov 20 '14 at 20:09
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Here is the wikipedia article which speaks of the ellipse which is tangent at the midpoints of a triangle: http://en.wikipedia.org/wiki/Steiner_inellipse In the end, it mentions that the above theorem as Marden's theorem: http://en.wikipedia.org/wiki/Marden's_theorem. In the following AMM article there is an elementary proof of the claim about the roots of the derivative of the polynomial: http://dankalman.net/AUhome/pdffiles/mardenAMM.pdf

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