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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.

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    Exhaustion arguments for deriving areas and volumes of plane/solid figures were just a jump, skip and hop from full blown integral calculus. – David H Apr 15 '13 at 19:21

3 Answers3

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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!

Dave L. Renfro
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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)

Lord_Farin
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xyzzyz
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The sum of the three angles in a triangle is variable, and not 180 degrees. If the triangle is on a sphere, the sum of the angles depends on its area, and this sum is more than 180 and less than 540 degrees. If the triangle lies on a hyperboloid surface, the angle sum also depends on its area, but now the sum is less than 180 degrees. On these surfaces, there is no similariry in the sense that similarity implies congruency. However, where the spherical surface "meets" the hyperboloid surface, so to speak, the sum of the angles becomes 180 degrees, where area is no longer related to angle sum and thus could be seen as a special case.

imranfat
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