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Take any closed, smooth, convex curve C, with a chord L on it. For any point X that divides L into segments of lengths p and q, let C' denote the locus of X as L moves all around C. (Note, L maintains its constant length throughout this motion.) Then the area between C and C' is equal to πpq.

Shaded area is πPQ:

 	 C---->    ...:,::::::::;;;;:::::::::::::::..                
               .:::,;;;;;;;;;;;:::;;;;;;;;;;;;;::::;\:.:::.             
          .:;::.:;;,;                              :;\...,;.:.          
      .::;:::.;;,                                     \;::;::;::.,      
     ,,,::.,:, <-----C'                               P\:,.:;....;,, 
   ,;...,:;:.                                           \ ;,...;;....,  
  ;..,,:.....;                                           \ ,::....;;,.. 
 ;;,::::::::::;.                  C' is locus of X       .X;,:......;;,.
...........::::;           as point O moves around C    ;..\........::::
::,,,........,;                                         :,..\..........;
 .,..,,:...;;;                                          .;:..\......... 
  ,,...:,,..,;.                                          ....Q\.,,,.:;  
   .,,....:;;..;                                          ,,,;.\..,:    
     .;..........                                      ,;;:.....\.-θ    
       ............                                  ,......,,;..O..\.............................      
           .:::,;;:;;;,,                         .:;;;,:::::,.          
                ::;:,;::::;;....,,,,,,,,,.....;;:::::::,,:              
                    ....;.:::::,;;,:::,::::::;,,......                          

Note that the result is entirely independent of C's shape. This was first proven by Holditch in 1858.


Source:

Anonymous. "Holditch's Theorem." Mathworld. Wolfram Research, 1999-2003. Online, available (http://mathworld.wolfram.com/HolditchsTheorem.html).

References:

Broman, A. "Holditch's Theorem." Math. Mag. Vol. 54, No. 1, pp. 99-108, 1981.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 103, 1991.
If anyone can explain to me how to prove this theorem, please /msg me.

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