4. Prove that there exists at least one line that simultaneously bisects both the area and the perimeter of a given convex polygon.
Problem 772
Official solution
4. a) First, let's prove that for every direction, there is exactly one line of that direction that bisects the given polygon (whether in terms of area or perimeter). For simplicity, let's consider horizontal lines. Translating the observed line from the top edge downwards, the area (or perimeter) of the upper part of the polygon increases, while that of the lower part decreases, so there is exactly one position of the line where both parts (upper and lower) are equal.
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