Without a doubt, my favourite theorem in all of mathematics — in the sense that it quite literally made me gape with astonishment when I first read it — is Marden’s Theorem. The result is named after Morris Marden, who publicized it in 1945 but actually attributes the result to Jörg Siebeck (1864), and provides a wonderfully simple geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative.
Theorem. Let p: ℂ → ℂ be any polynomial of degree three with complex coefficients such that the roots z1, z2 and z3 of p form a non-trivial triangle Δ in the complex plane ℂ. Then
- there is a unique ellipse E (known as the Steiner inellipse) that is tangent to all three sides of Δ at their midpoints;
- and, furthermore, the foci of the Steiner inellipse E are precisely the zeroes of the derivative p′ of p.
To me, that’s just incredible: you start with a cubic p, associate one obvious geometric object Δ with it by the “where are the zeroes?” operation, discover that there’s another unique geometric object E associated to Δ, and the two points (foci) that equivalently define E happen to be — no, don’t just happen to be, they have to be! — the answer to “what are the zeroes of p′?”
Footnote. It turns out that the Steiner ellipse E of Δ is also the John ellipse of Δ, meaning that it is the ellipse of maximal area that is contained within Δ. Higher-dimensional convex bodies also have associated John ellipsoids of maximal volume, but I don’t know if those ellipsoids have such nice algebraic relationships with the vertices of the containing bodies.