I'm interested in theorems which appear to have very few, if any hypotheses. Essentially a search for unexpected regularity or pattern in a relatively unstructured situation.


By "few hypotheses" I mean theorems which start "take any triangle", or "take any three circles". Similarly, the conclusion of the theorem ought to be really surprising. I know this is a little vague, but I've deliberately left it that way.


Perhaps my favourite here is Morley's theorem. This applies to *any* triangle, but has a very surprising conclusion. Contrast this with Pythagoras' theorem (needs a right angled triangle: too special!) or Viviani's Theorem (needs an equilateral triangle: too special!).


Can you help me gather a collection? I've made a very preliminary start here: <http://tube.geogebra.org/book/title/id/2673817>


Part of my underlying interest is in the aesthetic, and what professional mathematicians think is "significant", "surprising" or when exceptional cases mean the "take any .... except ..." means the theorem isn't so general after all.


Please don't be shy. I'd love to know what your favourites are. They don't have to be in geometry either.....


Chris Sangwin



A search for theorems which appear to have very few, if any hypotheses
Does the fundamental theorem of algebra count? 


Take *any* (non-constant) polynomial. 
Then it has a zero among the complex numbers.
I suppose this was quite surprising once complex numbers were new.


