Splitting fried eggs in a fair way

Question

This morning I was making a breakfast for two, part of which was frying 6 eggs (sunny side up) on a round-shaped pan. When it was ready I found it quite tricky to fairly split between the two: ideally I wanted to have a single cut that will go through the center, and leave 3 yolks on each side. That made me thinking of the following: it seems that if the pan was a perfect circle and yolks were dots, there will always be a line passing through the center that will leave exactly half of the dots in each half-circle (and that would apply to any even number of dots).

Questions:

1. Is my statement in italic true, and if it is, I'd be delighted to see an elegant proof of it (does not have to come from the school math).

2. I think this statement can be generalized as follows: we say that a connected Borel subset $A\subset\Bbb R^2$ satisfies the property E if for any even number of elements of $A$ there exists a line that splits $A$ into two sets of equal Lebesgue measure, each having an equal number of chosen elements. Does each convex set satisfy E, and if yes - are there non-convex sets with that property?

Answer 1

Ad 1: As exemplified by mathworker21's comment, the claim is false for specific yolk arrangements. However, let's add this mild condition:

No ray starting at the centre hits two or more yolks. (In particular, there is no yolk at the centre.)

Then the answer is "yes".

Given an angle $\phi$ such that the line in direction $\phi$ hits no yolk (which works for all but finitely many $\phi$), let $f(\phi)$ be the number of yolks to the left of direction $\phi$ minus the number on the right. Then $f$ is locally constant on its domain. Between adjacent intervals of the domain, $f$ may jump by $\pm 1$ because our line sweeps over a single yolk on one of its halves (or $f$ changes by $0$ if opposing yolks cancel). Note that $f(\phi)=-f(\phi+\pi)$. Hence if $f(\phi)\ne 0$ for any $\phi\in[0,\pi)$, it must reach $-f(\phi)$ in steps of size $\pm1$ as we sweep continuously from $\phi$ to $\phi+\pi$ and pass through $0$ on the way.

Note that a single ray wih two yolks on it can spoil the fun: Place all yolks in the same half of the pan in general position except that the middle two yolks are pushed on to a single ray. Then no cutting line with the desired properties exists.