Help wanted!

Chris Sangwin shared this question 6 years ago
Answered

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: https://www.geogebra.org/bo...


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


Chris

Comments (2)

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Do you mean like the Varignon Parallelogram.


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


Tony

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

Chris

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