Help wanted!

Chris Sangwin shared this question 6 years ago

Dear forum,

I'm posting to ask for your help. 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, together with their proofs? I've made a start here:

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


Comments (2)


Do you mean like the Varignon Parallelogram.

Do any of the theorems have famous names? You may have some already available within GeoGebraTube.



Yes Tony, I mean exactly like the Varignon parallelogram. I already included that as one of my 4! Thanks,


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