Theorems with almost no hypotheses
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?
Sommario
- Plane geometry
- Morley's Theorem
- Varignon's theorem
- Napoleon's theorem
- External tangents to three circles.
Morley's Theorem
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[b]Morley's Theorem [/b] Take any triangle ABC. "Trisect" the interior angles as shown. The intersection points of the adjacent trisectors form an equilateral triangle. |
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