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Marden's theorem: Given a triangle in the complex plane, a polynomial $p$ can be made from the vertices. The roots of derivative $p'$ give the foci of the Steiner inellipse, which is tangent to the midpoints of the original triangle.

That simple result ties together complex numbers, triangles, derivatives, polynomials and conic sections.

Euler's identity, $e^{i \pi} +1=0$, also covers numerous branches of mathematics.

What other simple results are similarly far-reaching?

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    [FLT](https://en.wikipedia.org/wiki/Fermat's_Last_Theorem) is far-reaching, involving very many things. However, only the statement is "simple". – 2017-01-09
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    Useful for the FLT, the [Riemann-Roch theorem](https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem#Riemann.E2.80.93Roch_theorem_for_algebraic_curves) has a version in Riemann surfaces and in finite fields, making a link between geometry, algebra and complex analysis. – 2017-01-10
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    [Monsky's theorem](https://en.wikipedia.org/wiki/Monsky's_theorem) looks like a good example. – 2017-01-10
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    there is also a Riemann-Roch theorem for combinatorial graphs and metric graphs, due to Baker and Norine https://mattbaker.blog/2013/10/18/riemann-roch-for-graphs-and-applications/ – 2017-10-25

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