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  "body": "\n## Statement\n\nLet $p$ be a polynomial of degree $3$ with complex coefficients whose three [[roots|019MA1VTR2]] $z_1, z_2, z_3$ are not collinear in the plane $\\C$.\nLet $T$ be the [[triangle|0CTNFGYMM2]] with vertices $z_1, z_2, z_3$.\nThen there is a unique [[ellipse|0VKCCEXBJ1]] inscribed in $T$ and tangent to its three sides at their midpoints.\nThe foci of this ellipse are the two roots of the derivative $p'$.\n\n## Remarks\n\nThe ellipse is the Steiner inellipse of $T$.\nIf $p$ is monic, then $p'(z) = (z - z_1)(z - z_2) + (z - z_2)(z - z_3) + (z - z_3)(z - z_1)$.\nOne proof computes the foci of the Steiner inellipse and checks, with the reflection property of the tangent lines of an ellipse, that they are the roots of this quadratic.\nFor an equilateral triangle the inellipse is the incircle, and for $p(z) = z^3 - 1$ its centre $0$ is the double root of $p'(z) = 3z^2$.\nExample: for $p(z) = z(z - 1)(z - i)$, one gets $p'(z) = 3z^2 - 2(1 + i)z + i$, whose roots are the foci of the inellipse of the triangle $0, 1, i$.\nThe theorem refines the Gauss–Lucas theorem for cubics: the critical points lie not only in the triangle but at the foci of a specific ellipse inside it.\n\n## Explanations\n\n### L1 human\n\nA cubic polynomial has three roots, which form a triangle in the plane.\nThere is exactly one ellipse that fits inside the triangle and touches each side at its midpoint.\nMarden's theorem says that the two foci of this ellipse are exactly the roots of the derivative.\nIt is a striking link between algebra and classical geometry.\n\n### L2 human\n\nThe roots of $p'$ are the [[critical points|0YTCMFB4HC]] of $p$, so the theorem locates the critical points of a cubic from its roots alone.\nIt is a classical result of the geometry of polynomials, which studies where the zeros of a polynomial and of its derivatives lie.\nTogether with [[Sendov's conjecture|101H2VSX2R]], it belongs to the study of how the zeros of $p'$ sit relative to the zeros of $p$.\n",
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