Pancakes again

Given three finite area, two-dimensional pancakes, prove that there exists a circle that cuts the area of each pancake into two equal pieces.

Originally posted by David113
Given three finite area, two-dimensional pancakes, prove that there exists a circle that cuts the area of each pancake into two equal pieces.

Can these pancakes be rearranged in the plane? Or do they have to stay where they lay on the griddle?

Originally posted by PBE6
Can these pancakes be rearranged in the plane? Or do they have to stay where they lay on the griddle?

The pancakes can't be rearranged.

Originally posted by David113
The pancakes can't be rearranged.

Just trying to think of counter-examples. What if you had three square pancakes lined up in a row? I can't think of a circle that would cut all three in half, unless you allow for an infinite radius, in which case the circle becomes a straight line. Is this allowed?

Originally posted by PBE6
Just trying to think of counter-examples. What if you had three square pancakes lined up in a row? I can't think of a circle that would cut all three in half, unless you allow for an infinite radius, in which case the circle becomes a straight line. Is this allowed?

I thought this, then thought if you get something that's so almost infinite radius this line is only slightly curved, then you can surely make it cut the three.

A pizza cutter? That would cut all three.

Are we using euclidean or Non-euclidean geometry?