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.

The Pascal theorem you mentioned can be easily derived from the Bézout theorem (see Frances Kirwan "Complex Algebraic Curves", Corollary 3.15), which in turn can be derived from the (very deep) theorem of Serre about global generation of tensor products of coherent sheaves with large enough tensor powers mentioned here, some facts about Euler-Poincaré characteristic and knowledge of the Picard group of a projective plane. See Arnaud Beauville "Complex Algebraic Surfaces", Example I.9 a)
