On 4 May, 20:43, Virgil <Vir...@[EMAIL PROTECTED]
> wrote:
> In article
> <16909701.1209927802890.JavaMail.jaka...@[EMAIL PROTECTED]
>,
>
> =A0amu <amu78...@[EMAIL PROTECTED]
> wrote:
> > Prove that there are at least one ways to cut the pizza and the
cheeseca=
ke in
> > halves of equal areas by one straight cut. (By pizzas and cheesecakes,
w=
e
> > mean compact connected subsets of R^2)
>
> Lemma: a compact connected subset of R^2 has a center of area.
What does "center of area" mean? Most lines through the centroid of a
triangle do not produce two halves of equal area. See
http://www.btinternet.com/~se16/js/halfarea.htm
In any direction there is a single line which bisects the area of the
pizza (just move the line continuously side-to-side across the pizza
to find it). As the direction changes continuously, this results in
an envelope of pizza-bisecting lines. If a partiuclar pizza-bisecting
line puts pro****tion x of the cheesecake on one side and 1-x on the
other, then rotating round the envelope by two right angles will put 1-
x and x of the cheesecake on each side of the same but reversed line.
Since this is a continuous process, somewhere in between on the
rotation will find a pizza-bisecting line which puts 1/2 and 1/2 of
the cheesecake on each side.
Usually called the pancake theorem (or ham sandwich theorem in three
dimensions)
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