Euclidean geometry in higher math? Often, we study a lot of euclidean geometry during high school. For example,Pascal's Theorem,Cross-ratio,Ceva's Theorem` and others. 
I am looking for instances of theorems encountered in Euclidean geometry which are actually special cases of deeper theorems in higher math. 
An example I can think of is the distance metric: http://en.wikipedia.org/wiki/Metric_(mathematics).
In particular, the condition $d(x,y)+d(y,z)\ge d(x,z)$ reminds me of the triangle inequality in the Euclidean plane.
I hope my question is not off-topic here.

1. Cavalieri's principle is a special case of Fubini's theorem.
2. The square-cube law can be considered as a special case of scaling principles based on fractal dimension notions.
3. Similar figures in geometry can be considered a special case of the notion of homeomorphic spaces in topology.
4. Euler's polyhedral formula is a special case of the general notion of the Euler characteristic of a chain complex.
5. The idea of a circumscribed circle of a polygon might be considered as foreshadowing the various covering results in analysis such as Vitali coverings and Besicovitch coverings. O-K, I'll admit that this one is a real leap!
