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.