No Problemmo #6

Geometry Level 5

There is a triangle EGO in the argand plane, whose vertices are the roots of a complex cubic polynomial. What is always true about the roots of the derivative of the polynomial?

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Lies on the line joining the circumcentre and orthocentre of EGO. Lies on the nine point circle of triangle EGO. Lies on the incircle of triangle EGO. Are the focii of an ellipse inscribed in the triangle EGO. None of these.

1 solution

Mark Hennings
Jan 6, 2018

This result is known as Marden's Theorem . If $F(z) = (z-u_1)(z-u_2)(z-u_3)$ is the cubic polynomial whose roots $u_1,u_2,u_3$ are the complex vertices of the triangle, then the roots of $F'(z)$ are the foci of the Steiner inellipse.

No Problemmo #6

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