Proof Without Words: Gregory’s Theorem

My favourite proof without words is the proof that the sum of the interior angles of any triangle is 180 degrees. Proof: draw an arbitrary triangle in chalk on the ground. Stand on one of the lines at one of the corners and note the direction you are facing. Now back up until you are at the corner of the triangle behind you. Rotate your self by shuffling inside the angle to the next line. Walk to the next corner, repeat. Walk (backwards this time) to the last corner, repeat. Now shuffle back to your starting position. You are facing exactly 180 degrees in the other direction!<p>Of course it is way better to just do this than write it down! But obviously I can&#x27;t do it on HN.

I haven&#x27;t previously come across the idea of trying to make wordless proofs. I guess this is the mathematical equivalent of code golf: a clever game for insiders to play, but basically the opposite of good practice from a readability perspective.

Here is another proof without words for the Pythagorean Theorem: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Pythagorean_theorem#&#x2F;media&#x2F;File:Pythagoras-proof-anim.svg" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Pythagorean_theorem#&#x2F;media&#x2F;Fil...</a>

Evidently Oliver Selfridge [1] said a puzzle based on deriving the radius like this once tripped Feynman up...<p>&quot;Puzzles from last week&quot; <a href="http:&#x2F;&#x2F;web.media.mit.edu&#x2F;~walter&#x2F;MAS-A12&#x2F;week11.html" rel="nofollow">http:&#x2F;&#x2F;web.media.mit.edu&#x2F;~walter&#x2F;MAS-A12&#x2F;week11.html</a><p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Oliver_Selfridge" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Oliver_Selfridge</a>

Hi there. Just a precision: the fact that the sum of the interior angles of a triangle is 180 degrees is not really provable, it is a postulate of the Euclidean geometry. It is an intuitive and accepted property, it reflects the fact that the geometric space usually considered is flat. Proving this statement would be possible provided that you change basic postulates. At that moment, it would become a proposition of these new mathematics and would certainly be provable starting from the new postulates. Changing these postulates is possible, that&#x27;s how we got the parabolic and hyperbolic geometries, see wiki!

great. now draw a few hundred more n. and don&#x27;t forget the min-max values.