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

π*pq*.
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.