One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal area and perimeter.
This problem can be resolved using intermediate value theorem. However the mathematical writing is a bit complicated. I found students gave erroneous arguments all over the place. Here is my question: Is there a clear, transparent way to prove this at a level the first time real analysis learner would understand? My own proof used some topology concepts, and I do not think I can explain it to the student really well. So I want to ask others for ideas.
A correct proof I have seen is to prove such a line dividing $\Omega$ into equal area exists for any angle $\theta$ by shifting the line. Similarly a line dividing $\Omega$ into equal perimeter exists. So if one take the equation for the difference of the two lines $h(\theta)=f(\theta)-g(\theta)$, there must exist some $\theta$ such that the two lines are the same since rotating changes the sign. But I feel this proof is not really clear. The difference of lines is ambiguous unless one interpret it as difference of equations, and the shifting process for the equation is not so clear. Even though I graded 5/5 for this student, I am wondering if there are clearer better proofs I can explain during office hours.