Puzzles/Riddles  - 
 
 
Puzzle: show the total area of the two semicircles is half the area of the large circle.

If you get stuck, go to +Alexander Bogomolny's wonderful website:

http://www.cut-the-knot.org/proofs/Semicircles.shtml

On top you'll get an applet that lets you slide the point where the semicircles touch - no matter where it is, the semicircles have the same total area!  Click "hint" and you'll get a hint.  If you're still stuck, scroll down and see a proof.

This puzzle is a lot harder than my other recent area puzzles.   Indeed, it seems this fact was proved only in 2011!  

• Andrew K. Jobbings, Two semicircles fill half a circle, The Mathematical Gazette 95 (Nov. 2011), 538-540.

I find that amazing, since people have been thinking about this stuff for millennia!  However, Andrew Jobbings is a genius when it comes to 'recreational mathematics' - by which I mean math that's not considered 'serious', which people do just because it's fun.  (This is a curious concept, now that I think about it.) 

Check out more of his stuff here:

http://www.arbelos.co.uk/papers.html

#geometry  
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7 comments
 
i didn't go to the site but looking at the drawing fresh I don't get it.  basically the larger radius squared would have to equal the sum of the other two squared. maybe i'll spend some time later and look into trying to show that that is true, or not true. certainly as the green one grows it eventually equals half the total space.
 
i have deduce some properties of these pythagorian circles. See my post above !
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