Marden's theorem

For a cubic polynomial with noncollinear complex roots, the zeros of its derivative are the foci of the Steiner inellipse of the triangle formed by those roots.

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Steiner inellipse: The unique ellipse tangent to all three sides of a triangle at their midpoints. It is the triangle's canonical inscribed ellipse whose foci Marden's theorem identifies.
Fundamental theorem of algebra: The theorem that every nonconstant complex polynomial has a complex root and factors completely into linear factors. It guarantees a cubic's three roots when multiplicities are counted.
Morris Marden: An American mathematician known for research on the geometry of zeros of polynomials. His name is attached to the theorem linking cubic critical points and the Steiner inellipse.
Equilateral triangle: A triangle whose three sides have equal length and whose angles are all 60 degrees. For equilateral roots, symmetry places the derivative zeros at the incenter and the ellipse center.
Gauss–Lucas theorem: The theorem that every zero of a polynomial's derivative lies in the convex hull of the polynomial's zeros. It gives a broad location constraint, while Marden identifies exact focal positions for cubics.
Focus (geometry): A distinguished point that helps define a conic section, such as an ellipse or parabola. The theorem locates the derivative zeros as the two foci of the inellipse.
Critical point (mathematics): A point where a function's derivative vanishes or is undefined. The derivative's zeros are the polynomial's critical points and become the ellipse's foci.
Jörgen Pedersen Gram: A Danish mathematician whose work included an early form of the result now known as Marden's theorem. The theorem's history reaches back to a related result published by Gram.
Isosceles triangle: A triangle with at least two equal sides. An isosceles root triangle gives a symmetric inellipse whose foci lie on its symmetry axis.
Rolle's theorem: A theorem stating that a differentiable function with equal endpoint values has a point with zero derivative between them. It is the real-variable ancestor of interlacing intuitions, but does not specify an ellipse.