The Pancake Theorem
Let me begin by saying that these theorems that will follow in this section of my blog will not be in any particular order, especially not in order of importance or preference. They are just popping up as I recall them or come across them.
Today's Theorem, the pancake theorem, is one that I just came across today. I don't remember if I have ever seen it before, but I have nominated it as my Theorem du jour.
The theorem says that the area of two plane pancakes (as in 2D, not plain pancakes), of arbitrary shape can be simultaneously bisected by a single straight-line cute of a knife. See the figure
Although, I don't have time now. I'll prove this in an update as well has give you the three dimensional version, known as the Ham Sandwich Theorem. . .
Now aren't you hungry?
UPDATE: Now here's a proof:
Lemma 1: Given a direction in the plane, there exists a line with that direction that bisects the figure.
Proof: This follows from the fact that if you create a function that measures the area to one side of the line, it is continuous with respect to the location. Thus, there must exist a line that bisects the area.
Now consider any circle that contains both pancakes and consider the diameter with direction for . By Lemma 1, we can make cuts in each pancake that are perpendicular to this diameter. Define and as the distance from the center to where these cuts cross the diameter. Then let .
Because, and are opposite directions we have that as well as . Thus, .
Now if , we are done and have bisected both pancakes. If then changes signs over and since is continuous, there exists some such that . Q.E.D.
As I mentioned there is an extension to this theorem for higher dimension. It is generally refered to as the Ham Sandwich theorem, obtaining its name for the fact that given three solids (e.g., two slices of bread and a slice of ham), there exists a cut that bisects all three simultaneously (giving each two people and exact equal share of all three).