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:


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:


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