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.